Simulating the Physical World: Hierarchical Modeling from Quantum Mechanics to Fluid Dynamics

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SIMULATING THE PHYSICAL WORLD

The simulation of physical systems requires a simpliﬁed, hierarchical approach, which models each level from the atomistic to the macroscopic scale. From quantum mechanics to ﬂuid dynamics, this book systematically treats the broad scope of computer modeling and simulations, describing the fundamental theory behind each level of approximation. Berendsen evaluates each stage in relation to their applications giving the reader insight into the possibilities and limitations of the models. Practical guidance for applications and sample programs in Python are provided. With a strong emphasis on molecular models in chemistry and biochemistry, this book will be suitable for advanced undergraduate and graduate courses on molecular modeling and simulation within physics, biophysics, physical chemistry and materials science. It will also be a useful reference to all those working in the ﬁeld. Additional resources for this title including solutions for instructors and programs are available online at www.cambridge.org/9780521835275. H e r m a n J . C . B e r e n d s e n is Emeritus Professor of Physical Chemistry at the University of Groningen. His research focuses on biomolecular modeling and computer simulations of complex systems. He has taught hierarchical modeling worldwide and is highly regarded in this ﬁeld.

SIMULATING THE PHYSICAL WORLD Hierarchical Modeling from Quantum Mechanics to Fluid Dynamics HERMAN J. C. BERENDSEN Emeritus Professor of Physical Chemistry, University of Groningen, the Netherlands

CAMBRIDGE UNIVERSITY PRESS

Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521835275 © H. J. C. Berendsen 2007 This publication is in copyright. Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published in print format 2007 eBook (EBL) ISBN-13 978-0-511-29491-4 ISBN-10 0-511-29491-3 eBook (EBL) hardback ISBN-13 978-0-521-83527-5 hardback ISBN-10 0-521-83527-5 paperback ISBN-13 978-0-521-54294-4 paperback ISBN-10 0-521-54294-4 Cambridge University Press has no responsibility for the persistence or accuracy of urls for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.

Contents

Preface Symbols, units and constants Part I

page xi xv

A Modeling Hierarchy for Simulations

1

1

Introduction 1.1 What is this book about? 1.2 A modeling hierarchy 1.3 Trajectories and distributions 1.4 Further reading

3 3 9 13 14

2

Quantum mechanics: principles and relativistic eﬀects 2.1 The wave character of particles 2.2 Non-relativistic single free particle 2.3 Relativistic energy relations for a free particle 2.4 Electrodynamic interactions 2.5 Fermions, bosons and the parity rule

19 19 23 25 31 36

3

From quantum to classical mechanics: when and how 3.1 Introduction 3.2 From quantum to classical dynamics 3.3 Path integral quantum mechanics 3.4 Quantum hydrodynamics 3.5 Quantum corrections to classical behavior

39 39 42 44 64 70

4

Quantum chemistry: solving the time-independent Schr¨ odinger equation 77 4.1 Introduction 77 4.2 Stationary solutions of the TDSE 78 4.3 The few-particle problem 79 4.4 The Born–Oppenheimer approximation 97 v

vi

Contents

4.5 4.6 4.7 4.8 4.9 4.10

The many-electron problem of quantum chemistry Hartree–Fock methods Density functional theory Excited-state quantum mechanics Approximate quantum methods Nuclear quantum states

98 99 102 105 106 107

5

Dynamics of mixed quantum/classical systems 5.1 Introduction 5.2 Quantum dynamics in a non-stationary potential 5.3 Embedding in a classical environment

109 109 114 129

6

Molecular dynamics 6.1 Introduction 6.2 Boundary conditions of the system 6.3 Force ﬁeld descriptions 6.4 Solving the equations of motion 6.5 Controlling the system 6.6 Replica exchange method 6.7 Applications of molecular dynamics

139 139 140 149 189 194 204 207

7

Free 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8

211 211 213 218 221 227 231 234 239

8

Stochastic dynamics: reducing degrees of freedom 8.1 Distinguishing relevant degrees of freedom 8.2 The generalized Langevin equation 8.3 The potential of mean force 8.4 Superatom approach 8.5 The ﬂuctuation–dissipation theorem 8.6 Langevin dynamics 8.7 Brownian dynamics 8.8 Probability distributions and Fokker–Planck equations 8.9 Smart Monte Carlo methods 8.10 How to obtain the friction tensor

energy, entropy and potential of mean force Introduction Free energy determination by spatial integration Thermodynamic potentials and particle insertion Free energy by perturbation and integration Free energy and potentials of mean force Reconstruction of free energy from PMF Methods to derive the potential of mean force Free energy from non-equilibrium processes

249 249 251 255 256 257 263 268 269 272 274

Contents

vii

9

Coarse graining from particles to ﬂuid dynamics 9.1 Introduction 9.2 The macroscopic equations of ﬂuid dynamics 9.3 Coarse graining in space 9.4 Conclusion

279 279 281 288 295

10

Mesoscopic continuum dynamics 10.1 Introduction 10.2 Connection to irreversible thermodynamics 10.3 The mean ﬁeld approach to the chemical potential

297 297 298 301

11

Dissipative particle dynamics 11.1 Representing continuum equations by particles 11.2 Prescribing ﬂuid parameters 11.3 Numerical solutions 11.4 Applications

305 307 308 309 309

Part II

313

Physical and Theoretical Concepts

12

Fourier transforms 12.1 Deﬁnitions and properties 12.2 Convolution and autocorrelation 12.3 Operators 12.4 Uncertainty relations 12.5 Examples of functions and transforms 12.6 Discrete Fourier transforms 12.7 Fast Fourier transforms 12.8 Autocorrelation and spectral density from FFT 12.9 Multidimensional Fourier transforms

315 315 316 317 318 320 323 324 325 331

13

Electromagnetism 13.1 Maxwell’s equation for vacuum 13.2 Maxwell’s equation for polarizable matter 13.3 Integrated form of Maxwell’s equations 13.4 Potentials 13.5 Waves 13.6 Energies 13.7 Quasi-stationary electrostatics 13.8 Multipole expansion 13.9 Potentials and ﬁelds in non-periodic systems 13.10 Potentials and ﬁelds in periodic systems of charges

335 335 336 337 337 338 339 340 353 362 362

viii

Contents

14

Vectors, operators and vector spaces 14.1 Introduction 14.2 Deﬁnitions 14.3 Hilbert spaces of wave functions 14.4 Operators in Hilbert space 14.5 Transformations of the basis set 14.6 Exponential operators and matrices 14.7 Equations of motion 14.8 The density matrix

379 379 380 381 382 384 385 390 392

15

Lagrangian and Hamiltonian mechanics 15.1 Introduction 15.2 Lagrangian mechanics 15.3 Hamiltonian mechanics 15.4 Cyclic coordinates 15.5 Coordinate transformations 15.6 Translation and rotation 15.7 Rigid body motion 15.8 Holonomic constraints

397 397 398 399 400 401 403 405 417

16

Review of thermodynamics 16.1 Introduction and history 16.2 Deﬁnitions 16.3 Thermodynamic equilibrium relations 16.4 The second law 16.5 Phase behavior 16.6 Activities and standard states 16.7 Reaction equilibria 16.8 Colligative properties 16.9 Tabulated thermodynamic quantities 16.10 Thermodynamics of irreversible processes

423 423 425 429 432 433 435 437 441 443 444

17

Review of statistical mechanics 17.1 Introduction 17.2 Ensembles and the postulates of statistical mechanics 17.3 Identiﬁcation of thermodynamical variables 17.4 Other ensembles 17.5 Fermi–Dirac, Bose–Einstein and Boltzmann statistics 17.6 The classical approximation 17.7 Pressure and virial 17.8 Liouville equations in phase space 17.9 Canonical distribution functions

453 453 454 457 459 463 472 479 492 497

Contents

18

19

ix

17.10 The generalized equipartition theorem

502

Linear response theory 18.1 Introduction 18.2 Linear response relations 18.3 Relation to time correlation functions 18.4 The Einstein relation 18.5 Non-equilibrium molecular dynamics

505 505 506 511 518 519

Splines for everything 19.1 Introduction 19.2 Cubic splines through points 19.3 Fitting splines 19.4 Fitting distribution functions 19.5 Splines for tabulation 19.6 Algorithms for spline interpolation 19.7 B-splines References Index

523 523 526 530 536 539 542 548 557 587

Preface

This book was conceived as a result of many years research with students and postdocs in molecular simulation, and shaped over several courses on the subject given at the University of Groningen, the Eidgen¨ ossische Technische Hochschule (ETH) in Z¨ urich, the University of Cambridge, UK, the University of Rome (La Sapienza), and the University of North Carolina at Chapel Hill, NC, USA. The leading theme has been the truly interdisciplinary character of molecular simulation: its gamma of methods and models encompasses the sciences ranging from advanced theoretical physics to very applied (bio)technology, and it attracts chemists and biologists with limited mathematical training as well as physicists, computer scientists and mathematicians. There is a clear hierarchy in models used for simulations, ranging from detailed (relativistic) quantum dynamics of particles, via a cascade of approximations, to the macroscopic behavior of complex systems. As the human brain cannot hold all the specialisms involved, many practical simulators specialize in their niche of interest, adopt – often unquestioned – the methods that are commonplace in their niche, read the literature selectively, and too often turn a blind eye on the limitations of their approaches. This book tries to connect the various disciplines and expand the horizon for each ﬁeld of application. The basic approach is a physical one, and an attempt is made to rationalize each necessary approximation in the light of the underlying physics. The necessary mathematics is not avoided, but hopefully remains accessible to a wide audience. It is at a level of abstraction that allows compact notation and concise reasoning, without the burden of excessive symbolism. The book consists of two parts: Part I follows the hierarchy of models for simulation from relativistic quantum mechanics to macroscopic ﬂuid dynamics; Part II reviews the necessary mathematical, physical and chemical concepts, which are meant to provide a common background of knowledge and notation. Some of these topics may be superﬂuous xi

xii

Preface

to physicists or mathematicians, others to chemists. The chapters of Part II could be useful in courses or for self-study for those who have missed certain topics in their education; for this purpose exercises are included. Answers and further information are available on the book’s website. The subjects treated in this book, and the depth to which they are explored, necessarily reﬂect the personal preference and experience of the author. Within this subjective selection the literature sources are restricted to the period before January 1, 2006. The overall emphasis is on simulation of large molecular systems, such as biomolecular systems where function is related to structure and dynamics. Such systems are in the middle of the hierarchy of models: very fast motions and the fate of electronically excited states require quantum-dynamical treatment, while the sheer size of the systems and the long time span of events often require severe approximations and coarse-grained approaches. Proper and eﬃcient sampling of the conﬁgurational space (e.g., in the prediction of protein folding and other rare events) poses special problems and requires innovative solutions. The fun of simulation methods is that they may use physically impossible pathways to reach physically possible states; thus they allow a range of innovative phantasies that are not available to experimental scientists. This book contains sample programs for educational purposes, but it contains no programs that are optimized to run on large or complex systems. For real applications that require molecular or stochastic dynamics or energy minimization, the reader is referred to the public-domain program suite Gromacs (http://www.gromacs.org), which has been described by Van der Spoel et al. (2005). Programming examples are given in Python, a public domain interpretative object-oriented language that is both simple and powerful. For those who are not familiar with Python, the example programs will still be intelligible, provided a few rules are understood: • Indentation is essential. Consecutive statements at the same indentation level are considered as a block, as if – in C – they were placed between curly brackets. • Python comes with many modules, which can be imported (or of which certain elements can be imported) into the main program. For example, after the statement import math the math module is accessible and the sine function is now known as math.sin. Alternatively, the sine function may be imported by from math import sin, after which it is known as sin. One may also import all the methods and attributes of the math module at once by the statement from math import ∗.

Preface

xiii

• Python variables need not be declared. Some programmers don’t like this feature as errors are more easily introduced, but it makes programs a lot shorter and easier to read. • Python knows several types of sequences or lists, which are very versatile (they may contain a mix of diﬀerent variable types) and can be manipulated. For example, if x = [1, 2, 3] then x[0] = 1, etc. (indexing starts at 0), and x[0 : 2] or x[: 2] will be the list [1, 2]. x + [4, 5] will concatenate x with [4, 5], resulting in the list [1, 2, 3, 4, 5]. x ∗ 2 will produce the list [1, 2, 3, 1, 2, 3]. A multidimensional list, as x = [[1, 2], [3, 4]] is accessed as x[i][j], e.g., x[0][1] = 2. The function range(3) will produce the list [0, 1, 2]. One can run over the elements of a list x by the statement for i in range(len(x)): . . . • The extra package numpy (numerical python) which is not included in the standard Python distribution, provides (multidimensional) arrays with ﬁxed size and with all elements of the same type, that have fast methods or functions like matrix multiplication, linear solver, etc. The easiest way to include numpy and – in addition – a large number of mathematical and statistical functions, is to install the package scipy (scientiﬁc python). The function arange acts like range, but deﬁnes an array. An array element is accessed as x[i, j]. Addition, multiplication etc. now work element-wise on arrays. The package deﬁnes the very useful universal functions that also work on arrays. For example, if x = array([1, 2, 3]), sin(x ∗ pi/2) will be array([1., 0., −1.]).

The reader who wishes to try out the sample programs, should install in this order: a recent version of Python (http://www.python.org), numpy and scipy (http://www.scipy.org) on his system. The use of the IDLE Python shell is recommended. For all sample programs in this book it is assumed that scipy has been imported: from scipy import * This imports universal functions as well, implying that functions like sin are known and need not be imported from the math module. The programs in this book can be downloaded from the Cambridge University Press website (http://www.cambridge.org/9780521835275) or from the author’s website (http://www.hjcb.nl). These sites also oﬀer additional Python modules that are useful in the context of this book: plotps for plotting data, producing postscript ﬁles, and physcon containing all relevant physical constants in SI

xiv

Preface

units. Instructions for the installation and use of Python are also given on the author’s website. This book could not have been written without the help of many former students and collaborators. It would never have been written without the stimulating scientiﬁc environment in the Chemistry Department of the University of Groningen, the superb guidance into computer simulation methods by Aneesur Rahman (1927–1987) in the early 1970s, the pioneering atmosphere of several interdisciplinary CECAM workshops, and the fruitful collaboration with Wilfred van Gunsteren between 1976 and 1992. Many ideas discussed in this book have originated from collaborations with colleagues, often at CECAM, postdocs and graduate students, of whom I can only mention a few here: Andrew McCammon, Jan Hermans, Giovanni Ciccotti, Jean-Paul Ryckaert, Alfredo DiNola, Ra´ ul Grigera, Johan Postma, Tjerk Straatsma, Bert Egberts, David van der Spoel, Henk Bekker, Peter Ahlstr¨ om, Siewert-Jan Marrink, Andrea Amadei, Janez Mavri, Bert de Groot, Steven Hayward, Alan Mark, Humberto Saint-Martin and Berk Hess. I thank Frans van Hoesel, Tsjerk Wassenaar, Farid Abraham, Alex de Vries, Agur Sevink and Florin Iancu for providing pictures. Finally, I thank my wife Lia for her endurance and support; to her I dedicate this book.

Symbols, units and constants

Symbols The typographic conventions and special symbols used in this book are listed in Table 1; Latin and Greek symbols are listed in Tables 2, 3, and 4. Symbols that are listed as vectors (bold italic, e.g., r) may occur in their roman italic version (r = |r|) signifying the norm (absolute value or magnitude) of the vector, or in their roman bold version (r) signifying a one-column matrix of vector components. The reader should be aware that occasionally the same symbol has a diﬀerent meaning when used in a diﬀerent context. Symbols that represent general quantities as a, unknowns as x, functions as f (x), or numbers as i, j, n are not listed. Units This book adopts the SI system of units (Table 5). The SI units (Syst`eme International d’Unit´es) were agreed in 1960 by the CGPM, the Conf´erence G´en´erale des Poids et Mesures. The CGPM is the general conference of countries that are members of the Metre Convention. Virtually every country in the world is a member or associate, including the USA, but not all member countries have strict laws enforcing the use of SI units in trade and commerce.1 Certain units that are (still) popular in the USA, such as inch (2.54 cm), ˚ Angstr¨ om (10−10 m), kcal (4.184 kJ), dyne (10−5 N), erg (10−7 J), bar (105 Pa), atm (101 325 Pa), electrostatic units, and Gauss units, in principle have no place in this book. Some of these, such as the ˚ A and bar, which are decimally related to SI units, will occasionally be used. Another exception that will occasionally be used is the still popular Debye for dipole moment (10−29 /2.997 924 58 Cm); the Debye relates decimally 1

A European Union directive on the enforcement of SI units, issued in 1979, has been incorporated in the national laws of most EU countries, including England in 1995.

xv

xvi

Symbols, units and constants

to the obsolete electrostatic units. Electrostatic and electromagnetic equations involve the vacuum permittivity (now called the electric constant) ε0 and vacuum permeability (now called the magnetic constant) μ0 ; the velocity of light does not enter explicitly into the equations connecting electric and magnetic quantities. The SI system is rationalized, meaning that electric and magnetic potentials, but also energies, ﬁelds and forces, are derived from their sources (charge density ρ, current density j) with a multiplicative factor 1/(4πε0 ), resp. μ0 /4π: ρ(r ) 1 Φ(r) = dr , (1) 4πε0 |r − r | μ0 j(r ) A(r) = dr , (2) 4π |r − r | while in diﬀerential form the 4π vanishes: div E = − div grad Φ = ρ/ε0 , curl B = curl curl A = μ0 j.

(3) (4)

In non-rationalized systems without a multiplicative factor in the integrated forms (as in the obsolete electrostatic and Gauss systems, but also in atomic units), an extra factor 4π occurs in the integrated forms: div E = 4πρ,

(5)

curl B = 4πj.

(6)

Consistent use of the SI system avoids ambiguities, especially in the use of electric and magnetic units, but the reader who has been educated with nonrationalized units (electrostatic and Gauss units) should not fall into one of the common traps. For example, the magnetic susceptibility χm , which is the ratio between induced magnetic polarization M (dipole moment per unit volume) and applied magnetic intensity H, is a dimensionless quantity, which nevertheless diﬀers by a factor of 4π between rationalized and nonrationalized systems of units. Another quantity that may cause confusion is the polarizability α, which is a tensor deﬁned by the relation μ = αE between induced dipole moment and electric ﬁeld. Its SI unit is F m2 , but its non-rationalized unit is a volume. To be able to compare α with a volume, the quantity α = α/(4πε0 ) may be deﬁned, the SI unit of which is m3 . Technical units are often based on the force exerted by standard gravity (9.806 65 m s−2 ) on a mass of a kilogram or a pound avoirdupois [lb = 0.453 592 37 kg (exact)], yielding a kilogramforce (kgf) = 9.806 65 N, or a poundforce (lbf) = 4.448 22 N. The US technical unit for pressure psi (pound

Symbols, units and constants

xvii

per square inch) amounts to 6894.76 Pa. Such non-SI units are avoided in this book. When dealing with electrons, atoms and molecules, SI units are not very practical. For treating quantum problems with electrons, as in quantum chemistry, atomic units (a.u.) are often used (see Table 7). In a.u. the electron mass and charge and Dirac’s constant all have the value 1. For treating molecules, a very convenient system of units, related to the SI system, uses nm for length, u (uniﬁed atomic mass unit) for mass, and ps for time. We call these molecular units (m.u.). Both systems are detailed below.

SI Units SI units are deﬁned by the basic units length, mass, time, electric current, thermodynamic temperature, quantity of matter and intensity of light. Units for angle and solid angle are the dimensionless radian and steradian. See Table 5 for the deﬁned SI units. All other units are derived from these basic units (Table 6). While the Syst`eme International also deﬁnes the mole (with unit mol ), being a number of entities (such as molecules) large enough to bring its total mass into the range of grams, one may express quantities of molecular size also per mole rather than per molecule. For macroscopic system sizes one then obtains more convenient numbers closer to unity. In chemical thermodynamics molar quantities are commonly used. Molar constants as the Faraday F (molar elementary charge), the gas constant R (molar Boltzmann constant) and the molar standard ideal gas volume Vm (273.15 K, 105 Pa) are speciﬁed in SI units (see Table 9).

Atomic units Atomic units (a.u.) are based on electron mass me = 1, Dirac’s constant = 1, elementary charge e = 1 and 4πε0 = 1. These choices determine the units of other quantities, such as 4πε0 2 , = 2 me e αme c me a20 (4πε0 )2 3 , = a.u. of time = 4 me e a.u. of velocity = /(me a0 ) = αc,

a.u. of length (Bohr radius) a0 =

(7) (8) (9)

xviii

Symbols, units and constants

a.u. of energy (hartree) Eh =

me e4 α2 c2 me = . (4πε0 )2 2 2

(10)

Here, α = e2 /(4πε0 c) is the dimensionless ﬁne-structure constant. The system is non-rationalized and in electromagnetic equations ε0 = 1/(4π) and μ0 = 4πα2 . The latter is equivalent to μ0 = 1/(ε0 c2 ), with both quantities expressed in a.u. Table 7 lists the values of the basic atomic units in terms of SI units. These units employ physical constants, which are not so constant as the name suggests; they depend on the deﬁnition of basic units and on the improving precision of measurements. The numbers given here refer to constants published in 2002 by CODATA (Mohr and Taylor, 2005). Standard errors in the last decimals are given between parentheses.

Molecular units Convenient units for molecular simulations are based on nm for length, u (uniﬁed atomic mass units) for mass, ps for time, and the elementary charge e for charge. The uniﬁed atomic mass unit is deﬁned as 1/12 of the mass of a 12 C atom, which makes 1 u equal to 1 gram divided by Avogadro’s number. The unit of energy now appears to be 1 kJ/mol = 1 u nm2 ps−2 . There is an electric factor fel = (4πε0 )−1 = 138.935 4574(14) kJ mol−1 nm e−2 when calculating energy and forces from charges, as in Vpot = fel q 2 /r. While these units are convenient, the unit of pressure (kJ mol−1 nm−3 ) becomes a bit awkward, being equal to 1.666 053 886(28) MPa or 16.66 . . . bar. Warning: One may not change kJ/mol into kcal/mol and nm into ˚ A (the usual units for some simulation packages) without punishment. When √ keeping the u for mass, the unit of time then becomes 0.1/ 4.184 ps = 48.888 821 . . . fs. Keeping the e for charge, the electric factor must be exA e−2 with a value of 332.063 7127(33). The unit of pressed in kcal mol−1 ˚ pressure becomes 69 707.6946(12) bar! These units also form a consistent system, but we do not recommend their use.

Physical constants In Table 9 some relevant physical constants are given in SI units; the values are those published by CODATA in 2002.2 The same constants are given in Table 10 in atomic and molecular units. Note that in the latter table 2

See Mohr and Taylor (2005) and http://physics.nist.gov/cuu/. A Python module containing a variety of physical constants, physcon.py, may be downloaded from this book’s or the author’s website.

Symbols, units and constants

xix

molar quantities are not listed: It does not make sense to list quantities in molecular-sized units per mole of material, because values in the order of 1023 would be obtained. The whole purpose of atomic and molecular units is to obtain “normal” values for atomic and molecular quantities.

xx

Symbols, units and constants

Table 1 Typographic conventions and special symbols Element

Example ∗

∗ ‡ hat overline dot bold italic (l.c.) bold italic (u.c.) bold roman (l.c.)

c ΔG‡ ˆ H u v˙ x r Q r

bold roman (u.c.) overline overline superscript T

Q u M bT AT H† df /dx ∂f /∂x D/Dt δA/δρ v·w v×w

superscript † d ∂ D δ centered dot × ∇ grad div grad

∇φ ∇·v ∇v

curl ∇2

∇×v ∇2 Φ

∇∇ tr calligraphic Z R C 1

∇∇Φ tr Q C

z z

Meaning complex conjugate c∗ = a − bi if c = a + bi transition state label operator (1) quantity per unit mass, (2) time average time derivative average over ensemble vector tensor of rank ≥ 2 one-column matrix, e.g., representing vector components matrix, e.g., representing tensor components quantity per unit mass multipole deﬁnition transpose of a column matrix (a row matrix) transpose of a rank-2 matrix (AT )ij = Aji ∗ Hermitian conjugate (H† )ij = Hji derivative function of f partial derivative Lagrangian derivative ∂/∂t + u · ∇ functional derivative dot product of two vectors vT w vector product of two vectors nabla vector operator (∂/∂x, ∂/∂y, ∂/∂z) gradient (∂φ/∂x, ∂φ/∂y, ∂φ/∂z) divergence (∂vx /∂x + ∂vy /∂y + ∂vz /∂z) gradient of a vector (tensor of rank 2) (∇v)xy = ∂vy /∂x curl v; (∇ × v)x = ∂vz /∂y − ∂vy /∂z Laplacian: nabla-square or Laplace operator (∂ 2 Φ/∂x2 + ∂ 2 Φ/∂y 2 + ∂ 2 Φ/∂z 2 ) Hessian (tensor) (∇∇Φ)xy = ∂ 2 Φ/∂x∂y trace of a matrix (sum of diagonal elements) set, domain or contour set of all integers (0, ±1, ±2, . . .) set of all real numbers set of all complex numbers real part of complex z imaginary part of complex z diagonal unit matrix or tensor

Symbols, units and constants

Table 2 List of lower case Latin symbols symbol a a0 c d e fel g h i j k k kB n m p p q [q] q r s t u u u v v w z z

meaning activity Bohr radius (1) speed of light, (2) concentration (molar density) inﬁnitesimal increment, as in dx (1) elementary charge, (2) number 2.1828 . . . electric factor (4πε0 )−1 metric tensor (1) Planck’s constant, (2) molar enthalpy Dirac’s constant (h/2π) √ −1 (j in Python programs) current density (1) rate constant, (2) harmonic force constant wave vector Boltzmann’s constant (1) total quantity of moles in a mixture, (2) number density mass of a particle (1) pressure, (2) momentum, (3) probability density (1) n-dimensional generalized momentum vector, (2) momentum vector mv (3D or 3N -D) (1) heat, mostly as dq, (2) generalized position, (3) charge [q0 , q1 , q2 , q3 ] = [q, Q] quaternions n-dimensional generalized position vector cartesian radius vector of point in space (3D or 3N -D) molar entropy time molar internal energy symbol for uniﬁed atomic mass unit (1/12 of mass 12 C atom) ﬂuid velocity vector (3D) molar volume cartesian velocity vector (3D or 3N -D) (1) probability density, (2) work, mostly as dw ionic charge in units of e point in phase space {q, p}

xxi

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Symbols, units and constants

Table 3 List of upper case Latin symbols Symbol A A B2 B D D E E F F G H H I J J K L L L M M N NA P P Q Q R R S dS S T T U V W W→ X

Meaning Helmholtz function or Helmholtz free energy vector potential second virial coeﬃcient magnetic ﬁeld vector diﬀusion coeﬃcient dielectric displacement vector energy electric ﬁeld vector Faraday constant (NA e = 96 485 C) force vector (1) Gibbs function or Gibbs free energy, (2) Green’s function (1) Hamiltonian, (2) enthalpy magnetic intensity moment of inertia tensor Jacobian of a transformation ﬂux density vector (quantity ﬂowing through unit area per unit time) kinetic energy Onsager coeﬃcients (1) Liouville operator, (2) Lagrangian angular momentum (1) total mass, (2) transport coeﬃcient (1) mass tensor, (2) multipole tensor (3) magnetic polarization (magnetic moment per unit volume) number of particles in system Avogadro’s number probability density (1) pressure tensor, (2) electric polarization (dipole moment per unit volume) canonical partition function quadrupole tensor gas constant (NA kB ) rotation matrix (1) entropy, (2) action surface element (vector perpendicular to surface) overlap matrix absolute temperature torque vector (1) internal energy, (2) interaction energy (1) volume, (2) potential energy (1) electromagnetic energy density transition probability thermodynamic driving force vector

Symbols, units and constants

xxiii

Table 4 List of Greek symbols Symbol α α β γ Γ δ Δ ε ε0 εr η ζ κ λ μ μ μ0 ν π Π Π ρ σ σ Σ τ τ φ Φ ψ Ψ χ χ2 Ξ ω ω Ω

Meaning (1) ﬁne structure constant, (2) thermal expansion coeﬃcient, (3) electric polarizability polarizability volume α/(4πε0 ) (1) compressibility, (2) (kB T )−1 (1) friction coeﬃcient as in v˙ = −γv, (2) activity coeﬃcient interfacial surface tension (1) delta function, (2) Kronecker delta: δij small increment, as in Δx (1) dielectric constant, (2) Lennard Jones energy parameter vacuum permittivity relative dielectric constant ε/ε0 viscosity coeﬃcient (1) bulk viscosity coeﬃcient, (2) friction coeﬃcient (1) inverse Debye length, (2) compressibility (1) wavelength, (2) heat conductivity coeﬃcient, (3) coupling parameter (1) thermodynamic potential, (2) magnetic permeability, (3) mean of distribution dipole moment vector vacuum permeability (1) frequency, (2) stoichiometric coeﬃcient number π = 3.1415 . . . product over terms momentum ﬂux density (1) mass density, (2) number density, (3) charge density (1) Lennard–Jones size parameter, (2) variance of distribution (3) irreversible entropy production per unit volume stress tensor sum over terms Poynting vector (wave energy ﬂux density) generalized time viscous stress tensor wave function (generally basis function) (1) wave function, (2) electric potential, (3) delta-response function wave function wave function, generally time dependent susceptibility: electric (χe ) or magnetic (χm ) chi-square probability function (1) grand-canonical partition function, (2) virial angular frequency (2πν) angular velocity vector microcanonical partition function

xxiv

Symbols, units and constants

Table 5 Deﬁned SI units Quantity

Name

Symbol

Deﬁnition (year adopted by CGPM)

length

meter

m

mass

kilogram

kg

time

second

s

current

ampere

A

temperature

kelvin

K

quantity

mole

mol

light intensity

candela

cd

distance traveled by light in vacuum in 1/299 792 458 s (1983) mass of international prototype kilogram in Paris (1889) duration of 9 192 631 770 periods of hyperﬁne transition in 133 Cs atoms [at rest at 0 K, in zero magnetic ﬁeld] (1967) current in two inﬁnitely long and thin conductors at 1 m distance that exert a mutual force of 2 × 10−7 N/m (1948) 1/273.16 of thermodynamic temperature of triple point of water (1967) quantity of matter with as many speciﬁed elementary entities as there are atoms in 0.012 kg pure 12 C (1971) intensity of light source emitting 1/683 W/sr radiation with frequency 540 × 1012 Hz (1979)

Table 6 Derived named SI units Quantity

Symbol

Name

Unit

planar angle solid angle frequency force pressure energy power charge electric potential capacity resistance conductance inductance magnetic ﬂux magnetic ﬁeld

α, . . . ω, Ω ν, f F p E, U, w P, W q, Q V, Φ C R G L Φ B

radian steradian hertz newton pascal joule watt coulomb volt farad ohm siemens henry weber tesla

rad (circle = 2π) sr (sphere= 4π) Hz = s−1 N = kg m s−2 Pa = N/m2 J = N m = kg m2 s−2 J s = kg m2 s−1 C = As V = J/C F = C/V Ω = V/A S = Ω−1 H = Wb/A Wb = V s T = Wb/m2

Symbols, units and constants

xxv

Table 7 Atomic units (a.u.) Quantity

Symbol

Value in SI unit

mass length time velocity energy

me a0 me a20 / αc 2 /(me a20 ) (Eh ) (hartree)

force charge current electric potential electric ﬁeld electric ﬁeld gradient dipole moment

Eh /a0 e a.u. a.u. a.u. a.u. a.u.

quadrupole moment electric polarizability α = α/(4πε0 )

a.u. a.u. a.u.

9.109 3826(16) × 10−31 kg 5.291 772 108(18) × 10−11 m 2.418 884 326505(16) × 10−17 s, 2.187 691 2633(73) × 106 m/s 4.359 744 17(75) × 10−18 J = 27.211 3845(23) eV = 2 625.499 63(45) kJ/mol = 627.509 47(11) kcal/mol 8.238 7225(14) × 10−8 N 1.602 176 53(14) × 10−19 C, 6.623 617 82(57) × 10−3 A 27.211 3845(23) V 5.142 206 42(44) × 1011 V/m 9.717 361 82(83) × 1021 V m−2 8.478 353 09(73) × 10−30 C m = 2.541 746 31(22) Debye 4.486 551 24(39) × 10−40 C m2 1.648 777 274(16) × 10−41 F m2 a30 = 1.481 847 114(15) × 10−31 m3

Table 8 Molecular units (m.u.) quantity

symbol

value in SI unit

mass length time velocity energy

u nm ps nm/ps kJ/mol

force charge current electric potential electric ﬁeld electric ﬁeld gradient dipole moment

kJ mol−1 nm−1 e e/ps kJ mol−1 e−1 kJ mol−1 e−1 nm−1 kJ mol−1 e−1 nm−2 e nm

quadrupole moment electric polarizability α = α/(4πε0 )

e nm2 e2 nm2 kJ−1 mol nm3

1.66053886(28) × 10−27 kg 1 × 10−9 m 1 × 10−12 s, 1000 m/s 1.660 538 86(28) × 10−21 J = 0.010 364 268 99(85) eV = 0.239 005 736 . . . kcal/mol 1.660 538 86(28) × 10−12 N 1.602 176 53(14) × 10−19 C, 1.602 176 53(14) × 10−7 A 0.010 364 268 99(85) V 1.036 426 899(85) × 107 V/m 1.036 426 899(85) × 1016 V m−2 1.602 176 53(14) × 10−28 C m = 48.032 0440(42) Debye 1.602 176 53(14) × 10−37 C m2 1.545 865 44(26) × 10−35 F m2 1 × 10−27 m3

xxvi

Symbols, units and constants

Table 9 Some physical constants in SI units (CODATA 2002) Constant magnetic constanta electric constantb electric factorc velocity of light gravitation constantd Planck constant Dirac constant electron mass elementary charge uniﬁed a.m.u.e proton mass neutron mass deuteron mass muon mass 1 H atom mass ﬁne-structure const. —, inverse Bohr radius Rydberg constantf Bohr magneton Boltzmann constant ideal gas volumeg Avogadro constant Faraday constant molar gas constant molar gas volumeh a b c d e f g h

Equivalent μ0 ε0 fel c G h me e u mp mn md mμ mH α α−1 a0 R∞ μB kB 0 vm NA F R Vm0

2 −1

(μ0 c ) (4πε0 )−1 def fund fund h/2π fund fund fund fund fund fund fund fund e2 /(2ε0 hc) 2ε0 hc/e2 /(αcme ) α2 me c/2h e/2me kB T 0 /p0 0.001 kg/u NA e NA kB RT 0 /p0

Value in SI units 4π × 10−7 (ex) N/A2 8.854 187 818... × 10−12 F/m 8.987 551 787... × 109 m/F 299 792 458(ex) m/s 6.6742(10) × 10−11 m3 kg−1 s−1 6.626 0693(11) × 10−34 J s 1.054 571 68(18) × 10−34 J s 9.109 3826(16) × 10−31 kg 1.602 176 53(14) × 10−19 C 1.66053886(28) × 10−27 kg 1.672 621 71(29) × 10−27 kg 1.674 927 28(29) × 10−27 kg 3.343 583 35(57) × 10−27 kg 1.883 531 40(33) × 10−28 kg 1.673 532 60(29) × 10−27 kg 7.297 352 568(24) × 10−3 137.035 999 11(46) 5.291 772 108(18) × 10−11 m 1.097 373 156 8525(73) × 107 m−1 9.274 009 49(80) × 10−24 J/T 1.380 6505(24) × 10−23 J/K 3.771 2467(66) × 10−26 m3 6.022 1415(10) × 1023 mol−1 96 485.3383(83) C/mol 8.314 472(15) J mol−1 K−1 22.710 981(40) × 10−3 m3 /mol

also called vacuum permeability. also called vacuum permittivity or vacuum dielectric constant. as in F = fel q1 q2 /r 2 . as in F = Gm1 m2 /r 2 . atomic mass unit, deﬁned as 1/12 of the mass of a 12 C atom very accurately known: relative uncertainty is 6.6 × 10−12 . volume per molecule of an ideal gas at a temperature of T 0 = 273.15 K and a pressure of p0 = 105 Pa. An alternative, but now outdated, standard pressure is 101 325 Pa. volume per mole of ideal gas under standard conditions; see previous note.

Symbols, units and constants

xxvii

Table 10 Physical constants in atomic units and “molecular units” Symbol μ0 ε0 fel c G h me e u mp mn md mμ mH α α−1 a0 R∞ μB kB 0 vm

Value in a.u.

Value in m.u. −4

6.691 762 564(44) × 10 1/(4π) 1(ex) 137.035 99911(46) 4.222 18(63) × 10−32 2π 1(ex) 1(ex) 1(ex) 1 822.888 484 93(80) 1 836.152 672 61(85) 1 838.683 6598(13) 3 670.482 9652(18) 206.768 2838(54) 1 837.152 645 89(85) 7.297 352 568(24) × 10−3 137.035 999 11(46) 1 (ex) 0.5(ex) 0.5(ex) 3.166 8154(55) × 10−6 254 496.34(44)

1.942 591 810(19) × 10−8 5.727 657 506(58) × 10−4 138.935 4574(14) 299 792.458(ex) 1.108 28(17) × 10−34 0.399 031 2716(27) 0.063 507 799 32(43) 5.485 799 0945(24) × 10−4 1(ex) 1(ex) 1.007 276 46688(13) 1.008 664 915 60(55) 2.013 553 212 70(35) 0.113 428 9264(30) 1.007 825 032 13(13) 7.297 352 568(24) × 10−3 137.035 999 11(46) 5.291 772 108(18) × 10−2 0.010 973 731 568 525(73) 57.883 818 04(39) 0.008 314 472(15) 37.712 467(66)

Part I A Modeling Hierarchy for Simulations

1 Introduction

1.1 What is this book about? 1.1.1 Simulation of real systems Computer simulations of real systems require a model of that reality. A model consists of both a representation of the system and a set of rules that describe the behavior of the system. For dynamical descriptions one needs in addition a speciﬁcation of the initial state of the system, and if the response to external inﬂuences is required, a speciﬁcation ofthe external inﬂuences. Both the model and the method of solution depend on the purpose of the simulation: they should be accurate and eﬃcient. The model should be chosen accordingly. For example, an accurate quantum-mechanical description of the behavior of a many-particle system is not eﬃcient for studying the ﬂow of air around a moving wing; on the other hand, the Navier–Stokes equations – eﬃcient for ﬂuid motion – cannot give an accurate description of the chemical reaction in an explosion motor. Accurate means that the simulation will reliably (within a required accuracy) predict the real behavior of the real system, and eﬃcient means “feasible with the available technical means.” This combination of requirements rules out a number of questions; whether a question is answerable by simulation depends on: • the state of theoretical development (models and methods of solution); • the computational capabilities; • the possibilities to implement the methods of solution in algorithms; • the possibilities to validate the model. Validation means the assessment of the accuracy of the model (compared to physical reality) by critical experimental tests. Validation is a crucial part of modeling. 3

4

Introduction

1.1.2 System limitation We limit ourselves to models of the real world around us. This is the realm of chemistry, biology and material sciences, and includes all industrial and practical applications. We do not include the formation of stars and galaxies (stellar dynamics) or the physical processes in hot plasma on the sun’s surface (astrophysics); neither do we include the properties and interactions of elementary particles (quantum chromodynamics) or processes in atomic nuclei or neutron stars. And, except for the purposes of validation and demonstration, we shall not consider unrealistic models that are only meant to test a theory. To summarize: we shall look at literally “down-to-earth” systems consisting of atoms and molecules under non-extreme conditions of pressure and temperature. This limits our discussion in practice to systems that are made up of interacting atomic nuclei, which are speciﬁed by their mass, charge and spin, electrons, and photons that carry the electromagnetic interactions between the nuclei and electrons. Occasionally we may wish to add gravitational interactions to the electromagnetic ones. The internal structure of atomic nuclei is of no consequence for the behavior of atoms and molecules (if we disregard radioactive decay): nuclei are so small with respect to the spatial spread of electrons that only their monopole properties as total charge and total mass are important. Nuclear excited states are so high in energy that they are not populated at reasonable temperatures. Only the spin degeneracy of the nuclear ground state plays a role when nuclear magnetic resonance is considered; in that case the nuclear magnetic dipole and electric quadrupole moment are important as well. In the normal range of temperatures this limitation implies a practical division between electrons on the one hand and nuclei on the other: while all particles obey the rules of quantum mechanics, the quantum character of electrons is essential but the behavior of nuclei approaches the classical limit. This distinction has far-reaching consequences, but it is rough and inaccurate. For example, protons are light enough to violate the classical rules. The validity of the classical limit will be discussed in detail in this book. 1.1.3 Sophistication versus brute force Our interest in real systems rather than simpliﬁed model systems is consequential for the kind of methods that can be used. Most real systems concern some kind of condensed phase: they (almost) never consist of isolated molecules and can (almost) never be simpliﬁed because of inherent

1.1 What is this book about?

5

symmetry. Interactions between particles can (almost) never be described by mathematically simple forms and often require numerical or tabulated descriptions. Realistic systems usually consist of a very large number of interacting particles, embedded in some kind of environment. Their behavior is (almost) always determined by statistical averages over ensembles consisting of elements with random character, as the random distribution of thermal kinetic energy over the available degrees of freedom. That is why statistical mechanics plays a crucial role in this book. The complexity of real systems prescribes the use of methods that are easily extendable to large systems with many degrees of freedom. Physical theories that apply to simple models only, will (almost) always be useless. Good examples are the very sophisticated statistical-mechanical theories for atomic and molecular ﬂuids, relating ﬂuid structural and dynamic behavior to interatomic interactions. Such theories work for atomic ﬂuids with simpliﬁed interactions, but become inaccurate and intractable for ﬂuids of polyatomic molecules or for interactions that have a complex form. While such theories thrived in the 1950s to 1970s, they have been superseded by accurate simulation methods, which are faster and easier to understand, while they predict liquid properties from interatomic interactions much more accurately. Thus sophistication has been superseded by brute force, much to the dismay of the sincere basic scientist. Many mathematical tricks that employ the simplicity of a toy model system cannot be used for large systems with realistic properties. In the example below the brute-force approach is applied to a problem that has a simple and elegant solution. To apply such a brute-force method to a simple problem seems outrageous and intellectually very dissatisfying. Nevertheless, the elegant solution cannot be readily extended to many particles or complicated interactions, while the brute-force method can. Thus not only sophistication in physics, but also in mathematics, is often replaced by brute force methods. There is an understandable resistance against this trend among well-trained mathematicians and physicists, while scientists with a less elaborate training in mathematics and physics welcome the opportunity to study complex systems in their ﬁeld of application. The ﬁeld of simulation has made theory much more widely applicable and has become accessible to a much wider range of scientists than before the “computer age.” Simulation has become a “third way” of doing science, not instead of, but in addition to theory and experimentation. There is a danger, however, that applied scientists will use “standard” simulation methods, or even worse use “black-box” software, without realizing on what assumptions the methods rest and what approximations are

6

Introduction V 6

2 V (r) = D 1 − e−a(r−b)

D

0 0

b

-r

Figure 1.1 Morse curve with a = 2/b (solid curve). Dotted curve: parabola with same curvature as Morse curve at r = b: V = Da2 (r − b)2 .

implied. This book is meant to provide the necessary scientiﬁc background and to promote awareness for the limitations and inaccuracies of simulating the “real world”. Example: An oscillating bond In this example we use brute-force simulation to attack a problem that could be approached analytically, albeit with great diﬃculty. Consider the classical bond length oscillation of a simple diatomic molecule, using the molecule hydrogen ﬂuoride (HF) as an example. In the simplest approximation the potential function is a parabola: V (r) = 12 k(r − b)2 ,

(1.1)

with r the H–F distance, k the force constant and b the equilibrium distance. A better description of the potential function is the Morse function (see Fig. 1.1) 2 V (r) = D 1 − e−a(r−b) ,

(1.2)

where D is the dissociation energy and a is a constant related to the steepness of the potential. The Morse curve is approximated near the minimum at r = b by a parabola with force constant k = 2Da2 . The Morse curve (Morse, 1929) is only a convenient analytical expression that has some essential features of a diatomic potential, including a fairly good agreement with vibration spectra of diatomic molecules, but there is no theoretical justiﬁcation for this particular form. In many occasions we may not even have an analytical form for the potential, but know the potential at a number of discrete points, e.g., from quantum-chemical calculations. In that case the best way to proceed is to construct the potential function from cubic spline interpolation of the computed points. Be-

1.1 What is this book about?

7

Table 1.1 Data for hydrogen ﬂuoride mass H mass F dissocation constant equilibrium bond length force constant

mH mF D b k

1.0079 18.9984 569.87 0.09169 5.82 × 105

u u kJ/mol nm kJ mol−1 nm−2

cause cubic splines (see Chapter 19) have continuous second derivatives, the forces will behave smoothly as they will have continuous ﬁrst derivatives everywhere. A little elementary mechanics shows that we can split oﬀ the translational motion of the molecule as a whole, and that – in the absence of rotational motion – the bond will vibrate according to the equation of motion: μ¨ r=−

dV , dr

(1.3)

where μ = mH mF /(mH + mF ) is the reduced mass of the two particles. When we start at time t = 0 with a displacement Δr and zero velocity, the solution for the harmonic oscillator is r(t) = b + Δr cos ωt, (1.4) with ω = k/μ. So the analytical solution is simple, and we do not need any numerical simulation to derive the frequency of the oscillator. For the Morse oscillator the solution is not as straightforward, although we can predict that it should look much like the harmonic oscillator with k = 2Da2 for small-amplitude vibrations. But we may expect anharmonic behavior for larger amplitudes. Now numerical simulation is the easiest way to derive the dynamics of the oscillator. For a spline-ﬁtted potential we must resort to numerical solutions. The extension to more complex problems, like the vibrations of a molecule consisting of several interconnected harmonic oscillators, is quite straightforward in a simulation program, while analytical solutions require sophisticated mathematical techniques. The reader is invited to write a simple molecular dynamics program that uses the following very general routine mdstep to perform one dynamics step with the velocity-Verlet algorithm (see Chapter 6, (6.83) on page 191). Deﬁne a function force(r) that provides an array of forces F , as well as the total potential energy V , given the coordinates r, both for the harmonic and the Morse potential. You may start with a one-dimensional version. Try out a few initial conditions and time steps and look for energy conservation and stability in long runs. As a rule of thumb: start with a time step such that the fastest oscillation period contains 50 steps (ﬁrst compute what the oscillation period will be). You may generate curves like those in Fig. 1.2. See what happens if you give the molecule a rotational velocity! In this case you of course need a two- or three-dimensional version. Keep to “molecular units”: mass: u, length: nm, time: ps, energy: kJ/mol. Use the data for hydrogen ﬂuoride from Table 1.1. The following function performs one velocity-Verlet time step of MD on a system of n particles, in m (one or more) dimensions. Given initial positions r, velocities v and forces F (at position r), each as arrays of shape (n, m), it returns r, v, F and

Introduction

H-F distance (nm)

8

0.13 0.12 0.11 0.1 0.09 0.08 5

10

15

20

25

30

35

40

time (fs) Figure 1.2 Oscillation of the HF bond length, simulated with the harmonic oscillator (solid curve) and the Morse curve (long dash), both with initial deviation from the equilibrium bond length of 0.01 nm, Dotted curve: Morse oscillator with initial deviation of 0.03 nm, showing increased anharmonic behavior. Note that the frequency of the Morse oscillator is lower than that of the harmonic oscillator. A time step of 0.2 fs was used; the harmonic oscillator simulation is indistinguishable from the analytical solution. the potential energy V one time step later. For convenience in programming, the inverse mass should be given as an array of the same shape (n, m) with repeats of the same mass for all m dimensions. In Python this n × m array invmass is easily generated from a one-dimensional array mass of arbitrary length n: invmass=reshape(repeat(1./mass,m),(alen(mass),m)), or equivalently invmass=reshape((1./mass).repeat(m),(alen(mass),m)) An external function force(r) must be provided that returns [F, V ], given r. V is not actually used in the time step; it may contain any property for further analysis, even as a list. python program 1.1 mdstep(invmass,r,v,F,force,delt) General velocity-Verlet Molecular Dynamics time step 01 02 03 04 05

def mdstep(invmass,r,v,F,force,delt): # invmass: inverse masses [array (n,m)] repeated over spatial dim. m # r,v,F: initial coordinates, velocities, forces [array (n,m)] # force(r): external routine returning [F,V] # delt: timestep

1.2 A modeling hierarchy 06 07 08 09 10 11

9

# returns [r,v,F,V] after step v=v+0.5*delt*invmass*F r=r+v*delt FV=force(r) v=v+0.5*delt*invmass*FV[0] return [r,v,FV[0],FV[1]]

Comments As mentioned in the Preface (page xiii), it is assumed that scipy has been imported. The initial values of r, v, F, V are valid at the time before the step, and normally available from the output of the previous step. To start the run, the routine force(r) must have been called once to initiate F . The returned values are valid at the end of the step. The arguments are not modiﬁed in place.

1.2 A modeling hierarchy The behavior of a system of particles is in principle described by the rules of relativistic quantum mechanics. This is – within the limitation of our system choices – the highest level of description. We shall call this level 1. All other levels of description, such as considering atoms and molecules instead of nuclei and electrons, classical dynamics instead of quantum dynamics, or continuous media instead of systems of particles, represent approximations to level 1. These approximations can be ordered in a hierarchical sense from ﬁne atomic detail to coarse macroscopic behavior. Every lower level loses detail and loses applicability or accuracy for a certain class of systems and questions, but gains applicability or eﬃciency for another class of systems and questions. The following scheme lists several levels in this hierarchy. LEVEL 1 relativistic quantum dynamics System Rules Atomic nuclei (mass, charge, spin), electrons (mass, charge, spin), photons (frequency)

Relativistic time-dependent quantum mechanics; Dirac’s equation; (quantum) electrodynamics

Approximation

No Go

Particle velocities small compared to velocity of light

Electrons close to heavy nuclei; hot plasmas

A A A

LEVEL 2 quantum dynamics System Rules Atomic nuclei, electrons, photons

A

Non-relativistic time-dependent Schr¨ odinger equation; timeindependent Schr¨ odinger equation; Maxwell equations

A A

10

Introduction

No Go Approximation Born–Oppenheimer approx.: electrons move much faster than nuclei

A A A

Electron dynamics (e.g., in semiconductors); fast electron transfer processes; dynamic behavior of excited states

A

A A

LEVEL 3 atomic quantum dynamics System Rules Atoms, ions, molecules, (photons)

Atoms move in eﬀective potential due to electrons; atoms may behave according to time-dependent Schr¨ odinger equation

No Go Approximation Atomic motion is classical A A A

Proton transfer; hydrogen and helium at low temperatures; fast reactions and highfrequency motions

A

A A

LEVEL 4 molecular dynamics System Rules Condensed matter: (macro)molecules, ﬂuids, solutions, liquid crystals, fast reactions

Classical mechanics (Newton’s equations); statistical mechanics; molecular dynamics

Approximation

No Go

Reduce number of degrees of freedom

Details of fast dynamics, transport properties

A A A

A

A A

LEVEL 5 generalized langevin dynamics on reduced system System Rules Condensed matter: large molecular aggregates, polymers, defects in solids, slow reactions

Approximation Neglect time correlation and/or spatial correlation in ﬂuctuations

Superatoms, reaction coordinates; averaging over local equilibrium, constraint dynamics, free energies and potentials of mean force.

No Go A A A

A

Correlations in motion, shorttime accuracy

A A

1.2 A modeling hierarchy

11

LEVEL 6 simple langevin dynamics System Rules “Slow” dynamic (non-equilibrium) processes and reactions

A

Approximation Neglect inertial terms: coarse graining in time

Accelerations given by systematic force, friction, and noise; Fokker– Planck equations

No Go Dynamic details A A A

A A

LEVEL 7 brownian dynamics System Rules Coarse-grained non-equilibrium processes; colloidal systems; polymer systems

Velocities given by force and friction, plus noise; Brownian (diﬀusive) dynamics; Onsager ﬂux/force relations

A

Approximation Reduce description to continuous densities of constituent species

No Go Details of particles A A A

A A

LEVEL 8 mesoscopic dynamics System Rules As for level 7: self-organizing systems; reactive non-equilibrium systems

Density description: mass conservation plus dynamic ﬂux equation, with noise.

Approximation

No Go

Average over “inﬁnite” number of particles

Spontaneous structure formation driven by ﬂuctuations

A A A

LEVEL 9 reactive fluid dynamics System Rules Non-equilibrium macroscopic mixture of diﬀerent species (as the atmosphere for weather forecasting

A

Energy, momentum and mass conservation; reactive ﬂuxes

A A

12

Introduction

Approximation

A

No Go

Reduce to one species with Newtonian viscosity

A A A

Reactive processes; Newtonian behavior

non

A A

LEVEL 10 fluid dynamics System Rules Non-equilibrium macroscopic ﬂuids: gases and liquids

A

Approximation Low ﬂuid velocities Reynolds number)

Energy, momentum and mass conservation; Navier–Stokes equation

No Go

(low

Turbulence A A A

A A

LEVEL 11 steady-flow fluid dynamics System Rules Non-equilibrium ﬂuids with laminar ﬂow

Simpliﬁed Navier–Stokes equation

From level 5 onward, not all atomic details are included in the system description: one speaks of coarse graining in space. From level 6 onward dynamic details on a short time scale are disregarded by coarse graining in time. In the last stages of this hierarchy (levels 8 to 11), the systems are not modeled by a set of particles, but rather by properties of a continuum. Equations describe the time evolution of the continuum properties. Usually such equations are solved on a spatial grid using ﬁnite diﬀerence or ﬁnite elements methods for discretizing the continuum equations. A diﬀerent approach is the use of particles to represent the continuum equations, called dissipative particle dynamics (DPD). The particles are given the proper interactions representing the correct physical properties that ﬁgure as parameters in the continuum equations. Note that we have considered dynamical properties at all levels. Not all questions we endeavor to answer involve dynamic aspects, such as the prediction of static equilibrium properties (e.g., the binding constant of a ligand to a macromolecule or a solid surface). For such static questions the answers may be found by sampling methods, such as Monte Carlo simulations, that generate a representative statistical ensemble of system conﬁgurations rather than a trajectory in time. The ensemble generation makes use of random

1.3 Trajectories and distributions

13

displacements, followed by an acceptance or rejection based on a probabilistic criterion that ensures detailed balance between any pair of conﬁgurations: the ratio of forward and backward transition probabilities is made equal to the ratio of the required probabilities of the two conﬁgurations. In this book the emphasis will be on dynamic methods; details on Monte Carlo methods can be found in Allen and Tildesley (1987) or Frenkel and Smit (2002) for chemically oriented applications and in Binder and Heermann (2002) or Landau and Binder (2005) for physically oriented applications.

1.3 Trajectories and distributions Dynamic simulations of many-particle systems contain ﬂuctuations or stochastic elements, either due to the irrelevant particular choice of initial conditions (as the exact initial positions and velocities of particles in a classical simulation or the speciﬁcation of the initial wave function in a quantumdynamical simulation), or due to the “noise” added in the method of solution (as in Langevin dynamics where a stochastic force is added to replace forces due to degrees of freedom that are not explicitly represented). Fluctuations are implicit in the dynamic models up to and including level 8. The precise details of a particular trajectory of the particles have no relevance for the problem we wish to solve. What we need is always an average over many trajectories, or at least an average property, such as the average or the variance of a single observable or a correlation function, over one long trajectory. In fact, an individual trajectory may even have chaotic properties: two trajectories with slightly diﬀerent initial conditions may deviate drastically after a suﬃciently long time. However, the average behavior is deterministic for most physical systems of interest. Instead of generating distribution functions and correlation functions from trajectories, we can also try to deﬁne equations, such as the Fokker–Planck equation, for the distribution functions (probability densities) or correlation functions themselves. Often the latter is very much more diﬃcult than generating the distribution functions from particular trajectories. An exception is the generation of equilibrium distributions, for which Monte Carlo methods are available that circumvent the necessity to solve speciﬁc equations for the distribution functions. Thus the simulation of trajectories is often the most eﬃcient – if not the only possible – way to generate the desired average properties. While the notion of a trajectory as the time evolution of positions and velocities of all particles in the system is quite valid and clear in classical mechanics, there is no such notion in quantum mechanics. The description

14

Introduction

of a system in terms of a wave function Ψ is by itself a description in terms of a probability density: Ψ∗ Ψ(r 1 , . . . , r n , t) is the probability density that the particles 1, . . . , n are at positions r 1 , . . . , r n at time t. Even if the initial state is precisely deﬁned by a sharp wave function, the wave function evolves under the quantum-dynamical equations to yield a probability distribution rather than a precise trajectory. From the wave function evolution expectation values (i.e., average properties over a probability distribution) of physical observables can be obtained by the laws of quantum mechanics, but the wave function cannot be interpreted as the (unmeasurable) property of a single particle. Such a description ﬁts in well with equations for the evolution of probability distributions in classical systems, but it is not compatible with descriptions in terms of classical trajectories. This fundamental diﬀerence in interpretation lies at the basis of the diﬃculties we encounter if we attempt to use a hybrid quantum/classical description of a complex system. If we insist on a trajectory description, the quantum-dynamical description should be reformulated by some kind of contraction and sampling to yield trajectories that have the same statistical properties as prescribed by the quantum evolution. It is for the same reason of incompatibility of quantum descriptions and trajectories that quantum corrections to classical trajectories cannot be unequivocally deﬁned, while quantum corrections to equilibrium probability distributions can be systematically derived.

1.4 Further reading While Part I treats most of the theoretical models behind simulation and Part II provides a fair amount of background knowledge, the interested reader may feel the need to consult standard texts on further background material, or consult books on aspects of simulation and modeling that are not treated in this book. The following literature may be helpful. 1 S. Gasiorowicz, Quantum Physics (2003) is a readable, over 30 years old but updated, textbook on quantum physics with a discussion of the limits of classical physics. 2 L. I. Schiﬀ, Quantum Mechanics (1968). A compact classic textbook, slightly above the level of Gasiorowicz. 3 E. Merzbacher, Quantum Mechanics (1998) is another classic textbook with a complete coverage of the main topics. 4 L. D. Landau and E.M. Lifshitz, Quantum Mechanics (Non-relativis-

1.4 Further reading

5

6

7

8 9

10

11

12 13 14

15

15

tic Theory) (1981). This is one volume in the excellent series “Course of Theoretical Physics.” Its level is advanced and sophisticated. P. A. M. Dirac, The Principles of Quantum Mechanics (1958). By one of the founders of quantum mechanics: advisable reading only for the dedicated student. F. S. Levin, An Introduction to Quantum Theory (2002) introduces principles and methods of basic quantum physics at great length. It has a part on “complex systems” that does not go far beyond twoelectron atoms. A. Szabo and N. S. Ostlund, Modern Quantum Chemistry (1982) is a rather complete textbook on quantum chemistry, entirely devoted to the solution of the time-independent Schr¨ odinger equation for molecules. R. McWeeny, Methods of Molecular Quantum Mechanics (1992) is the classical text on quantum chemistry. R. G. Parr and W. Yang, Density Functional Theory (1989). An early, and one of the few books on the still-developing area of densityfunctional theory. F. Jensen, Introduction to Computational Chemistry (2006). First published in 1999, this is a modern comprehensive survey of methods in computational chemistry including a range of ab initio and semiempirical quantum chemistry methods, but also molecular mechanics and dynamics. H. Goldstein, Classical Mechanics (1980) is the classical text and reference book on mechanics. The revised third edition (Goldstein et al., 2002) has an additional chapter on chaos, as well as other extensions, at the expense of details that were present in the ﬁrst two editions. L. D. Landau and E. M. Lifshitz, Mechanics (1982). Not as complete as Goldstein, but superb in its development of the theory. L. D. Landau and E. M. Lifshitz, Statistical Physics (1996). Basic text for statistical mechanics. K. Huang, Statistical Mechanics (2nd edn, 1987). Statistical mechanics textbook from a physical point of view, written before the age of computer simulation. T. L. Hill, Statistical Mechanics (1956). A classic and complete, but now somewhat outdated, statistical mechanics textbook with due attention to chemical applications. Written before the age of computer simulation.

16

Introduction

16 D. A. McQuarrie, Statistical Mechanics (1973) is a high-quality textbook, covering both physical and chemical applications. 17 M. Toda, R. Kubo and N. Saito, Statistical Physics. I. Equilibrium Statistical Mechanics (1983) and R. Kubo, M. Toda and N. Hashitsume Statistical Physics. II. Nonequilibrium Statistical Mechanics (1985) emphasize physical principles and applications. These texts were originally published in Japanese in 1978. Volume II in particular is a good reference for linear response theory, both quantummechanical and classical, to which Kubo has contributed signiﬁcantly. It describes the connection between correlation functions and macroscopic relaxation. Not recommended for chemists. 18 D. Chandler, Introduction to Modern Statistical Mechanics (1987). A basic statistical mechanics textbook emphasizing ﬂuids, phase transitions and reactions, written in the age of computer simulations. 19 B. Widom, Statistical Mechanics, A Concise Introduction for Chemists (2002) is what it says: an introduction for chemists. It is wellwritten, but does not reach the level to treat the wonderful inventions in computer simulations, such as particle insertion methods, for which the author is famous. 20 M. P. Allen and D. J. Tildesley, Computer Simulation of Liquids (1987). A practical guide to molecular dynamics simulations with emphasis on the methods of solution rather than the basic underlying theory. 21 D. Frenkel and B. Smit, Understanding Molecular Simulation (2002). A modern, instructive, and readable book on the principles and practice of Monte Carlo and molecular dynamics simulations. 22 D. P. Landau and K. Binder, A Guide to Monte Carlo Simulations in Statistical Physics (2005). This book provides a detailed guide to Monte Carlo methods with applications in many ﬁelds, from quantum systems to polymers. 23 N. G. van Kampen, Stochastic Processes in Physics andChemistry (1981) gives a very precise and critical account of the use of stochastic and Fokker–Planck type equations in (mostly) physics and (a bit of) chemistry. 24 H. Risken, The Fokker–Planck equation (1989) treats the evolution of probability densities. 25 C. W. Gardiner, Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences (1990) is a reference book for modern developments in stochastic dynamics. It treats the relations between stochastic equations and Fokker–Planck equations.

1.4 Further reading

17

26 M. Doi and S. F. Edwards, The Theory of Polymer Dynamics (1986) is the already classic introduction to mesoscopic treatment of polymers. 27 L. D. Landau and E. M. Lifshitz, Fluid Mechanics (1987) is an excellent account of the physics behind the equations of ﬂuid dynamics. 28 T. Pang, Computational Physics (2006). First published in 1997, this is a modern and versatile treatise on methods in computational physics, covering a wide range of applications. The emphasis is on the computational aspects of the methods of solution, not on the physics behind the models. 29 F. J. Vesely, Computational Physics, An Introduction (2nd ed., 2001) is an easily digestable treatment of computational problems in physics, with emphasis on mathematical and computational methodology rather than on the physics behind the equations. 30 M. Griebel, S. Knapek, G. Zumbusch and A. Caglar, Numerische Simulation in der Molek¨ uldynamik (2003) gives many advanced details on methods and algorithms for dynamic simulation with particles. The emphasis is on computational methods including parallelization techniques; programs in C are included. Sorry for some readers: the text is in German. 31 D. Rapaport, The Art of Molecular Dynamics Simulation (2004) is the second, reworked edition of a detailed, and readable, account of classical molecular dynamics methods and applications. 32 M. M. Woolfson and G. J. Pert, An Introduction to Computer Simulation (1999) is not on models but on methods, from solving partial diﬀerential equations to particle simulation, with accessible mathematics. 33 A. R. Leach, Molecular Modelling, Principles and Applications (1996) aims at the simulation of molecular systems leading up to drug discovery. Starting with quantum chemistry, the book decribes energy minimization, molecular dynamics and Monte Carlo methods in detail. 34 C. J. Cramer, Essentials of Computational Chemistry (2004) is the second edition of a detailed textbook of modern computational chemistry including quantum methods, simulation, optimization and reaction dynamics.

2 Quantum mechanics: principles and relativistic eﬀects

Readers who are not sensitive to the beauty of science can skip this entire chapter, as nothing is said that will help substantially to facilitate the solution of practical problems! LEVEL 1 relativistic quantum dynamics System Rules Atomic nuclei (mass, charge, spin), electrons (mass, charge, spin), photons (frequency)

Relativistic time-dependent quantum mechanics; Dirac’s equation; (quantum) electrodynamics

Approximation

No Go

Particle velocities small compared to velocity of light

Electrons close to heavy nuclei; hot plasmas

A A A

A

A A

LEVEL 2 quantum dynamics System Rules Atomic nuclei, electrons, photons

Non-relativistic time-dependent Schr¨ odinger equation; timeindependent Schr¨ odinger equation; Maxwell equations

2.1 The wave character of particles Textbooks on quantum mechanics abound, but this is not one of them. Therefore, an introduction to quantum mechanics is only given here as a guideline to the approximations that follow. Our intention is neither to be complete nor to be rigorous. Our aim is to show the beauty and simplicity of the basic quantum theory; relativistic quantum theory comprises such 19

20

Quantum mechanics: principles and relativistic eﬀects

subtleties as electron spin, spin-orbit and magnetic interactions in a natural way. For practical reasons we must make approximations, but by descending down the hierarchy of theoretical models, we unfortunately lose the beauty of the higher-order theories. Already acquired gems, such as electron spin, must be re-introduced at the lower level in an ad hoc fashion, thus muting their brilliance. Without going into the historical development of quantum mechanics, let us put two classes of observations at the heart of quantum theory: • Particles (such as electrons in beams) show diﬀraction behavior as if they are waves. The wavelength λ appears to be related to the momentum p = mv of the particle by λ = h/p, where h is Planck’s constant. If we deﬁne k as the wave vector in the direction of the velocity of the particle and with absolute value k = 2π/λ, then p = k

(2.1)

(with = h/2π) is a fundamental relation between the momentum of a particle and its wave vector. • Electromagnetic waves (such as monochromatic light) appear to consist of packages of energy of magnitude hν, where ν is the frequency of the (monochromatic) wave, or ω, where ω = 2πν is the angular frequency of the wave. Assuming that particles have a wave character, we may generalize this to identify the frequency of the wave with the energy of the particle: E = ω.

(2.2)

Let us further deﬁne a wave function Ψ(r, t) that describes the wave. A homogeneous plane wave, propagating in the direction of k with a phase velocity ω/k is described by Ψ(r, t) = c exp[i(k · r − ωt)] where c is a complex constant, the absolute value of which is the amplitude of the wave, while its argument deﬁnes the phase of the wave. The use of complex numbers is a matter of convenience (restriction to real numbers would require two amplitudes, one for the sine and one for the cosine constituents of the wave; restriction to the absolute value would not enable us to describe interference phenomena). In general, a particle may be described by a superposition of many (a continuum of) waves of diﬀerent wave vector and frequency: Ψ(r, t) = dk dωG(k, ω) exp[i(k · r − ωt)], (2.3)

2.1 The wave character of particles

21

where G is a distribution function of the wave amplitude in k, ω space. Here we recognize that Ψ(r, t) and G(k, ω) are each other’s Fourier transform, although the sign conventions for the spatial and temporal transforms differ. (See Chapter 12 for details on Fourier transforms.) Of course, the transform can also be limited to the spatial variable only, yielding a timedependent distribution in k-space (note that in this case we introduce a factor of (2π)−3/2 for symmetry reasons): −3/2 Ψ(r, t) = (2π) dk g(k, t) exp[i(k · r)]. (2.4) The inverse transform is −3/2

g(k, t) = (2π)

dr Ψ(r, t) exp(−ik · r).

(2.5)

The next crucial step is one of interpretation: we interpret Ψ∗ Ψ(r, t) as the probability density that the particle is at r at time t. Therefore we require for a particle with continuous existence the probability density to be normalized at all times: (2.6) dr Ψ∗ Ψ(r, t) = 1, where the integration is over all space. Likewise g ∗ g is the probability density in k-space; the normalization of g ∗ g is automatically satisﬁed (see Chapter 12): (2.7) dk g ∗ g(k, t) = 1 The expectation value, indicated by triangular brackets, of an observable f (r), which is a function of space only, then is (2.8) f (r)(t) = dr Ψ∗ Ψ(r, t)f (r) and likewise the expectation value of a function of k only is given by f (k)(t) = dk g ∗ g(k, t)f (k). (2.9) If we apply these equations to deﬁne the expectation values of the variances of one coordinate x and its conjugate k = kx : σx2 = (x − x)2 ,

(2.10)

σk2 = (k − k)2 ,

(2.11)

22

Quantum mechanics: principles and relativistic eﬀects

we can show (see Chapter 12) that σx σk ≥ 12 ,

(2.12)

which shows that two conjugated variables as x and k (that appear as product ikx in the exponent of the Fourier transform) cannot be simultaneously sharp. This is Heisenberg’s uncertainty relation, which also applies to t and ω. Only for the Gaussian function exp(−αx2 ) the product of variances reaches the minimal value. As shown in Chapter 12, averages over k and powers of k can be rewritten in terms of the spatial wave function Ψ: (2.13) k(t) = dr Ψ∗ (−i∇)Ψ(r, t), dr Ψ∗ (−∇2 )Ψ(r, t). (2.14) k 2 (t) = Thus, the expectation of some observable A, being either a function of r only, or being proportional to k or to k 2 , can be obtained from ˆ A(t) = dr Ψ∗ (r, t)AΨ(r, t), (2.15) where Aˆ is an operator acting on Ψ, and ˆ = −i∇, k kˆ2 = −∇2 .

(2.16) (2.17)

Similarly (but with opposite sign due to the opposite sign in ωt), the expectation value of the angular frequency ω is found from equation by using the operator ω ˆ=i

∂ . ∂t

(2.18)

The identiﬁcations p = k (2.1) and E = ω (2.2) allow the following operator deﬁnitions: ˆ = −i∇, p ˆ p2 = −2 ∇2 , ˆ = i ∂ . E ∂t

(2.19) (2.20) (2.21)

From these relations and expression of the energy as a function of momenta and positions, the equations of motion for the wave function follow.

2.2 Non-relativistic single free particle

23

2.2 Non-relativistic single free particle In principle we need the relativistic relations between energy, momentum and external ﬁelds, but for clarity we shall ﬁrst look at the simple nonrelativistic case of a single particle in one dimension without external interactions. This will allow us to look at some basic propagation properties of wave functions. Using the relation p2 E= , (2.22) 2m then (2.21) and (2.20) give the following equations of motion for the wave function: 2 ∂ 2 Ψ ∂Ψ(x, t) =− , (2.23) i ∂t 2m ∂x2 or i ∂ 2 Ψ ∂Ψ = . (2.24) ∂t 2m ∂x2 This is in fact the time-dependent Schr¨ odinger equation. This equation looks much like Fick’s diﬀusion equation, with the diﬀerence that the diﬀusion constant is now imaginary (or, equivalently, that the diﬀusion takes place in imaginary time). If you don’t know what that means, you are in good company. If we choose an initial wave function Ψ(x, 0), with Fourier transform g(k, 0), then the solution of (2.24) is simply ∞ 1 dk g(k, 0) exp[ikx − iω(k)t], (2.25) Ψ(x, t) = √ 2π −∞ k 2 (2.26) ω(k) = 2m The angular frequency corresponds to the energy: E = ω =

(k)2 p2 = , 2m 2m

(2.27)

as it should. If ω had been just proportional to k (and not to k 2 ) then (2.25) would represent a wave packet traveling at constant velocity without any change in the form of the packet. But because of the k 2 -dependence the wave packet slowly broadens as it proceeds in time. Let us assume that g(k, 0) is a narrow distribution around a constant k0 , and write k 2 = k02 + 2k0 Δk + (Δk)2 , k0 = mv.

(2.28) (2.29)

24

Quantum mechanics: principles and relativistic eﬀects

In these terms the wave function can be written as 1 Ψ(x, t) = √ exp[ik0 (x − 12 vt)] 2π ∞ dΔk g(Δk, t) exp[iΔk(x − vt)], =

(2.30)

−∞

g(k, t) = g(k, 0) exp[−i

(Δk)2 t ]. 2m

(2.31)

The factor in front of the integral is a time-dependent phase factor that is irrelevant for the shape of the density distribution since it cancels in Ψ∗ Ψ. The packet shape (in space) depends on x = x − vt and thus the packet travels with the group velocity v = dω/dk. However, the packet changes shape with time. In fact, the package will always broaden unless it represents a stationary state (as a standing wave), but the latter requires an external conﬁning potential. Let us take a Gaussian packet with initial variance (of Ψ∗ |Ψ) of σ02 and with velocity v (i.e., k = k0 ) as an example. Its initial description (disregarding normalizing factors) is

x2 (2.32) Ψ(x, 0) ∝ exp − 2 + ik0 x , 4σ0 g(k, 0) ∝ exp[−σ02 (Δk)2 ].

(2.33)

The wave function Ψ(x = x − vt, t) is, apart from the phase factor, equal to the inverse Fourier transform in Δk of g(k, t) of (2.31):

t g(k, t) ∝ exp − σ02 + i (2.34) (Δk)2 , 2m which works out to

x2 . Ψ(x, t) ∝ exp − 4(σ02 + it/2m)

(2.35)

By evaluating Ψ∗ Ψ, we see that a Gaussian density is obtained with a variance σ(t) that changes in time according to 2 t2 2 2 . (2.36) σ (t) = σ0 1 + 4m2 σ04 The narrower the initial package, the faster it will spread. Although this seems counterintuitive if we think of particles, we should remember that the wave function is related to the probability of ﬁnding the particle at a certain place at a certain time, which is all the knowledge we can possess. If the

2.3 Relativistic energy relations for a free particle

25

initial wave function is narrow in space, its momentum distribution is broad; this implies a larger uncertainty in position when time proceeds. Only free particles broaden beyond measure; in the presence of conﬁning potentials the behavior is quite diﬀerent: stationary states with ﬁnite width emerge. Because the packet becomes broader in space, it seems that the Heisenberg uncertainty relation would predict that it therefore becomes sharper in momentum distribution. This, however, is an erroneous conclusion: the broadening term is imaginary, and Ψ is not a pure real Gaussian; therefore the relation σx σk = 1/2 is not valid for t > 0. In fact, the width in k-space remains the same.

2.3 Relativistic energy relations for a free particle The relation between energy and momentum (for a free particle) that we used in the previous section (2.22) is incorrect for velocities that approach the speed of light. In non-relativistic physics we assume that the laws of physics are invariant for a translation of the spatial and time origins of our coordinate system and also for a rotation of the coordinate system; this leads to fundamental conservation laws for momentum, energy, and angular momentum, respectively.1 In the theory of special relativity the additional basic assumption is that the laws of physics, including the velocity of light, are also invariant if we transform our coordinate system to one moving at a constant speed with respect to the original one. Where for normal rotations in 3D-space we require that the square of length elements (dr)2 is invariant, the requirement of the constant speed of light implies that for transformations to a moving frame (c dτ )2 = (c dt)2 − (dr)2 is invariant. For 1 + 1 dimensions where we transform from (x, t) to x , t in a frame moving with velocity v, this leads to the Lorentz transformation x x γ −γv/c , (2.37) = ct −γv/c γ ct where γ=

1 1 − v 2 /c2

.

(2.38)

In Minkovsky space of 1 + 3 dimensions (ct, x, y, z) = (ct, r) vectors are fourvectors vμ = (v0 , v)(μ = 0, 1, 2, 3) and we deﬁne the scalar or inner product 1

Landau and Lifschitz (1982) give a lucid derivation of these laws.

26

Quantum mechanics: principles and relativistic eﬀects

of two four-vectors as def

vμ wμ = v0 w0 − v1 w1 − v2 w2 − v3 w3 = v0 w0 − v · w.

(2.39)

The notation vμ wμ uses the Einstein summation convention ( 3μ=0 over repeating indices is assumed, taking the signs into account as in (2.39)).2 The square magnitude or length of a four-vector is the scalar product with itself; note that such a square length may be positive or negative. Lorentz transformations are all transformations in Minkowski space that leave dxμ dxμ = (c dτ )2 invariant; they of course include all space-like rotations for which dτ = 0. Vectors that represent physical quantities are invariant for Lorentz transformations, and hence their scalar products and square magnitudes are constants. Without any derivation, we list a number of relevant physical four-vectors, as they are deﬁned in relativistic mechanics: • coordinates: xμ = (ct, r); • wave vector: kμ = (ω/c, k); • velocity: uμ = (γc, γv); • momentum: pμ = muμ = (γmc, γmv). Here m is the (rest) mass of the particle. The ﬁrst component of the momentum four-vector is identiﬁed with the energy E/c, so that E = γmc2 . Note the following constant square lengths: u μ u μ = c2 , E2 pμ pμ = 2 − p2 = m2 c2 , c

(2.40)

E 2 = m2 c4 + p2 c2 .

(2.42)

(2.41)

or

This is the relation between energy and momentum that we are looking for. From the quadratic form it is immediately clear that E will have equivalent positive and negative solutions, one set around +mc2 and the other set around −mc2 . Only the ﬁrst set corresponds to the solutions of the nonrelativistic equation. 2

We use subscripts exclusively and do not use general tensor notation which distinguishes covariant and contravariant vectors and uses a metric tensor to deﬁne vector products. We note that the “Einstein summation convention” in non-relativistic contexts, for example in matrix multiplication, is meant to be simply a summation over repeated indices.

2.3 Relativistic energy relations for a free particle

27

Now identifying E with i∂/∂t and p with −i∇, we obtain the Klein– Gordon equation

∂2 mc 2 − 2 2 + ∇2 − Ψ = 0. (2.43) c ∂t This equation has the right relativistic symmetry (which the Schr¨ odinger equation does not have), but unfortunately no solutions with real scalar densities Ψ∗ Ψ exist. Dirac devised an ingeneous way to linearize (2.42). Let us ﬁrst consider the case of one spatial dimension, where motion is allowed only in the xdirection, and angular momentum cannot exist. Instead of taking a square root of (2.42), which would involve the square root of the operator pˆ, one can devise a two-dimensional matrix equation which in fact equals a set of equations with multiple solutions: i

∂Ψ ˆ = c(αˆ p + βmc)Ψ = HΨ, ∂t

(2.44)

where Ψ is a two-component vector, and α and β are dimensionless Hermitian 2 × 2 matrices, chosen such that (2.42) is satisﬁed for all solutions of (2.44): p2 + m2 c2 )1. (αˆ p + βmc)2 = (ˆ

(2.45)

p + β 2 m2 c2 = (ˆ p2 + m2 c2 )1, α2 pˆ2 + (αβ + βα)mcˆ

(2.46)

This implies that

or α2 = 1,

β 2 = 1,

αβ + βα = 0.

(2.47)

In other words, α and β are Hermitian, anticommuting, and unitary matrices.3 The trivial solutions of the ﬁrst two equations: α = ±1 and/or β = ±1 do not satisfy the third equation. There are many solutions to all three equations (2.47). In fact, when a matrix pair α, β forms a solution, the matrix pair UαU† , UβU† , constructed by a unitary transformation U, forms a solution as well. A simple choice is 1 0 0 1 . (2.48) ; β= α= 0 −1 1 0 3

Hermitian (α† = α) because the eigenvalues must be real, anticommuting because αβ + βα = 0, unitary because α2 = α† α = 1.

28

Quantum mechanics: principles and relativistic eﬀects

Inserting this choice into (2.44) yields the following matrix diﬀerential equation: ∂ ΨL ΨL mc pˆ =c . (2.49) i ΨS pˆ −mc dt ΨS We see that in a coordinate frame moving with the particle (p = 0) there are two solutions: ΨL corresponding to particles (electrons) with positive energy E = mc2 ; and ΨS corresponding to antiparticles (positrons) with negative energy E = −mc2 . With non-relativistic velocities p mc, the wave function ΨS mixes slightly in with the particle wave function ΨL (hence the subscripts L for “large” and S for “small” when we consider particles). The eigenfunctions of the Hamiltonian matrix are ˆ = ±c(m2 c2 + pˆ2 )1/2 , H

(2.50)

which gives, after expanding the square root to ﬁrst order in powers of p/mc, the particle solution ∂Ψ pˆ2 i ≈ (mc2 + )Ψ, (2.51) ∂t 2m in which we recognize the Schr¨ odinger equation for a free particle, with an extra constant, and irrelevant, zero-energy term mc2 . In the case of three spatial dimensions, there are three α-matrices for each of the spatial components; i.e., they form a vector α of three matrices αx , αy , αz . The simplest solution now requires four dimensions, and Ψ becomes a four-dimensional vector. The Dirac equation now reads ∂Ψ ˆ ˆ + βmc)Ψ = HΨ, = c(α · p (2.52) ∂t where αx , αy , αz and β are mutually anti-commuting 4 × 4 matrices with their squares equal to the unit matrix. One choice of solutions is: 0 σ , (2.53) α= σ 0 i

σx =

0 1 1 0

,

σy =

0 −i i 0

,

σz =

1 0 0 −1

,

(2.54)

while β is a diagonal matrix {1, 1, −1, −1} that separates two solutions around +mc2 from two solutions around −mc2 . The wave function now also has four components, which refer to the two sets of solutions (electrons and positrons) each with two spin states. Thus spin is automatically introduced; it gives rise to an angular momentum S and an extra quantum number S = 1/2. By properly incorporating electromagnetic interactions, the small

2.3 Relativistic energy relations for a free particle

29

spin-orbit interaction, arising from magnetic coupling between the electron spin S and angular orbital momentum L, is included in the solution of the Dirac equation. This term makes it impossible to exactly separate the spin and orbital momenta; in fact there is one quantum number for the total angular momentum. Let us now look at the relativistic eﬀects viewed as a perturbation of the non-relativistic Schr¨ odinger equation. We may ﬁrst remark that spin can be separately and ad hoc introduced into the non-relativistic case as a new degree of freedom with two states. Each electron spin has associated with it an angular momentum S and a magnetic moment μ = −γe S, where γe is the electron’s gyromagnetic ratio. The spin-orbit interaction term can then be computed from the classical interaction of the electron magnetic moment with the magnetic ﬁeld that arises at the electron due to its orbital motion around a charged nucleus. The relativistic eﬀects arising from the high velocity of the electron can be estimated from a Taylor expansion of the positive solution of (2.42): E = m2 c2 + p2 , (2.55) c p2 p4 − + · · · , (2.56) E = mc2 1 + 2m2 c2 8m4 c4 p4 p2 = mc2 + − + ··· (2.57) 2m 8m3 c2 The ﬁrst term is an irrelevant zero-point energy, the second term gives us the non-relativistic Schr¨ odinger equation, and the third term gives a relativistic correction. Let us estimate its magnitude by a classical argument. Assume that the electron is in a circular orbital at a distance r to a nucleus with charge Ze. From the balance between the nuclear attraction and the centrifugal force we conclude that p2 = −2mE,

(2.58)

where E is the (negative) total energy of the electron, not including the term mc2 (this also follows from the virial equation valid for a central Coulombic ﬁeld: Epot = −2Ekin , or E = −Ekin ). For the expectation value of the ﬁrst relativistic correction we ﬁnd a lower bound p4 p2 2 E2 ≥ = . (2.59) 3 2 3 2 8m c 8m c 2mc2 The correction is most important for 1s-electrons near highly charged nuclei; since −E is proportional to Z 2 , the correction is proportional to Z 4 . For

30

Quantum mechanics: principles and relativistic eﬀects

the hydrogen atom E = −13.6 eV while mc2 = 511 keV and hence the correction is 0.18 meV or 17 J/mol; for germanium (charge 32) the eﬀect is expected to be a million times larger and be in the tens of MJ/mol. Thus the eﬀect is not at all negligible and a relativistic treatment for the inner shells of heavy atoms is mandatory. For molecules with ﬁrst-row atoms the relativistic correction to the total energy is still large (-146 kJ/mol for H2 O), but the eﬀects on binding energies and on equilibrium geometry are small (dissociation energy of H2 O into atoms: -1.6 kJ/mol, equilibrium OH distance: -0.003 pm, equilibrium angle: -0.08 deg).4 In addition to the spin and energetic eﬀects, the 1s-wave functions contract and become “smaller”; higher s-wave functions also contract because they remain orthogonal to the 1s-functions. Because the contracted s-electrons oﬀer a better shielding of the nuclear charge, orbitals with higher angular momentum tend to expand. The eﬀect on outer shell behavior is a secondary eﬀect of the perturbation of inner shells: therefore, for quantum treatments that represent inner shells by eﬀective core potentials, as in most practical applications of density functional theory, the relativistic corrections can be well accounted for in the core potentials without the need for relativistic treatment of the outer shell electrons. Relativistic eﬀects show up most clearly in the properties of heavy atoms, such as gold (atom number 79) and mercury (80). The fact that gold has its typical color, in contrast to silver (47) which has a comparable electron conﬁguration, arises from the relatively high energy of the highest occupied d-orbital (due to the expansion of 5d3/2 -orbital in combination with a high spin-orbit coupling) and the relatively low energy of the s-electrons in the conduction band (due to contraction of the 6s-orbitals), thus allowing light absorption in the visible region of the spectrum. The fact that mercury is a liquid (in contrast to cadmium (48), which has a comparable electron conﬁguration) arises from the contraction of the 6s-orbitals, which are doubly occupied and so localized and “buried” in the electronic structure that they contribute little to the conduction band. Mercury atoms therefore resemble noble gas atoms with weak interatomic interactions. Because the enthalpy of fusion (2.3 kJ/mol) is low, the melting point (234 K) is low. For cadmium the heat of fusion is 6.2 kJ/mol and the melting point is 594 K. For the same reason mercury is a much poorer conductor (by a factor of 14) than cadmium. For further reading on this subject the reader is referred to Norrby (1991) and Pyykk¨ o (1988).

4

Jensen (1999), p. 216.

2.4 Electrodynamic interactions

31

2.4 Electrodynamic interactions ¿From the relation E = p2 /2m and the correspondence relations between energy or momentum and time or space derivatives we derived the nonrelativistic Schr¨ odinger equation for a non-interacting particle (2.24). How is this equation modiﬁed if the particle moves in an external potential? In general, what we need is the operator form of the Hamiltonian H, which for most cases is equivalent to the total kinetic plus potential energy. When the potential energy in an external ﬁeld is a function V (r) of the coordinates only,such as produced by a stationary electric potential, it is simply added to the kinetic energy: i

∂Ψ 2 2 =− ∇ Ψ + V (r)Ψ. ∂t 2m

(2.60)

In fact, electrons feel the environment through electromagnetic interactions, in general with both an electric and a magnetic component. If the electric ﬁeld is not stationary, there is in principle always a magnetic component. As we shall see, the magnetic component acts through the vector potential that modiﬁes the momentum of the particle. See Chapter 13 for the basic elements of electromagnetism. In order to derive the proper form of the electromagnetic interaction of a particle with charge q and mass m, we must derive the generalized momentum in the presence of a ﬁeld. This is done by the Lagrangian formalism of mechanics, which is reviewed in Chapter 15. The Lagrangian L(r, v) is deﬁned as T −V , where T is the kinetic energy and V is the potential energy. In the case of an electromagnetic interaction, the electrical potential energy is modiﬁed with a velocity-dependent term −qA · v, where A is the vector potential related to the magnetic ﬁeld B by B = curl A,

(2.61)

in a form which is invariant under a Lorentz transformation: V (r, v) = qφ − qA · v.

(2.62)

Thus the Lagrangian becomes L(r, v) = 12 mv 2 − qφ + qA · v.

(2.63)

The reader should verify that with this Lagrangian the Euler–Lagrange equations of motion for the components of coordinates and velocities d ∂L ∂L = (2.64) dt ∂vi ∂xi

32

Quantum mechanics: principles and relativistic eﬀects

lead to the common Lorentz equation for the acceleration of a charge q in an electromagnetic ﬁeld mv˙ = q(E + v × B),

(2.65)

where def

E = −∇φ −

∂A ∂t

(2.66)

(see Chapter 13). The generalized momentum components pi are deﬁned as (see Chapter 15) ∂L pi = , (2.67) ∂vi and hence p = mv + qA,

(2.68)

or 1 (p − qA). (2.69) m For the Schr¨ odinger equation we need the Hamiltonian H, which is deﬁned as (see Chapter 15) v=

1 (p − qA)2 + qφ. (2.70) 2m Thus the non-relativistic Schr¨ odinger equation of a particle with charge q and mass m, in the presence of an electromagnetic ﬁeld, is

2 2 ∂Ψ iqA ˆ = − i = HΨ + qφ(r) Ψ. (2.71) ∇− ∂t 2m def

H = p·v−L=

Being non-relativistic, this description ignores the magnetic eﬀects of spin and orbital momentum, i.e., both the spin-Zeeman term and the spin-orbit interaction, which must be added ad hoc if required.5 The magnetic ﬁeld component of the interaction between nuclei and electrons or electrons mutually is generally ignored so that these interactions are described by the pure Coulomb term which depends only on coordinates and not on velocities. If we also ignore magnetic interactions with external ﬁelds (A = 0), we obtain for a N -particle system with masses mi and 5

The Dirac equation (2.52) in the presence of an external ﬁeld (A, φ) has the form: i

∂Ψ ˆ = [cα · (ˆ p − qA) + βmc2 + qφ1]Ψ = HΨ. ∂t

This equation naturally leads to both orbital and spin Zeeman interaction with a magnetic ﬁeld and to spin-orbit interaction. See Jensen (1999).

2.4 Electrodynamic interactions

33

charges qi the time-dependent Schr¨ odinger equation for the wave function Ψ(r 1 , . . . , r N , t): i

∂Ψ ˆ = HΨ ∂t 2 = [− ∇2 + 2mi i

1 1 2 4πε0

qi qj i,j rij

+ Vext (r 1 , . . . , r N , t)]Ψ,(2.72)

i

where the 1/2 in the mutual Coulomb term corrects for double counting in the sum, the prime on the sum means exclusion of i = j, and the last term, if applicable, represents the energy in an external ﬁeld. Let us ﬁnally derive simpliﬁed expressions in the case of external electromagnetic ﬁelds. If the external ﬁeld is a “slow” electric ﬁeld, (2.72) suﬃces. If the external ﬁeld is either a “fast” electric ﬁeld (that has an associated magnetic ﬁeld) or includes a separate magnetic ﬁeld, the nabla operators should be modiﬁed as in (2.71) to include the vector potential: iq A(r i ). (2.73) For simplicity we now drop the particle index i, but note that for ﬁnal results summation over particles is required. Realizing that ∇i → ∇i −

∇ · (AΨ) = (∇ · A)Ψ + A · (∇Ψ), the kinetic energy term reduces to 2 2iq q2 iq iq A · ∇ − 2 A2 . ∇ − A = ∇2 − (∇ · A) −

(2.74)

(2.75)

Let us consider two examples: a stationary homogeneous magnetic ﬁeld B and an electromagnetic plane wave. 2.4.1 Homogeneous external magnetic ﬁeld Consider a constant and homogeneous magnetic ﬁeld B and let us ﬁnd a solution A(r) for the equation B = curl A. There are many solutions (because any gradient ﬁeld may be added) and we choose one for which ∇ · A = 0 (the Lorentz convention for a stationary ﬁeld, see Chapter 13): A(r) = 12 B × r.

(2.76)

The reader should check that this choice gives the proper magnetic ﬁeld while the divergence vanishes. The remaining terms in (2.75) are a linear term in A: A · ∇ = 12 (B × r) · ∇ = 12 B · (r × ∇),

(2.77)

34

Quantum mechanics: principles and relativistic eﬀects

which gives a term in the Hamiltonian that represents the Zeeman interaction of the magnetic ﬁeld with the orbital magnetic moment: ˆ ˆ zeeman = e B · L, (2.78) H 2m ˆ = r×p ˆ = −i r × ∇, L (2.79) ˆ is the dimensionless orbital angular momentum operator, and a where L quadratic term in A that is related to magnetic susceptibility.6 The Zeeman interaction can be considered as the energy −μ·B of a dipole in a ﬁeld; hence the (orbital) magnetic dipole operator equals ˆ =− μ

e ˆ ˆ L = −μB L, 2m

(2.80)

where μB = e/2m is the Bohr magneton. In the presence of spin this modiﬁes to ˆ, ˆ = −gμB J μ (2.81) where ˆ =L ˆ + S, ˆ J

(2.82)

and g is the Lande g-factor, which equals 1 for pure orbital contributions, 2.0023 for pure single electron-spin contributions, and other values for mixed ˆ is characterized by a quantum states. The total angular momentum J number J and, if the spin-orbit coupling is small, there are also meaningful quantum numbers L and S for orbital and spin angular momentum. The g-factor then is approximately given by g =1+

J(J + 1) + S(S + 1) − L(L + 1) . 2J(J + 1)

(2.83)

2.4.2 Electromagnetic plane wave In the case of perturbation by an electromagnetic wave (such as absorption of light) we describe for simplicity the electromagnetic ﬁeld by a linearly polarized monochromatic plane wave in the direction k (see Chapter 13): E = E 0 exp[i(k · r − ωt)], 1 k ×E , B = c k ω = kc. 6

For details see Jensen (1999).

(2.84) (2.85) (2.86)

2.4 Electrodynamic interactions

35

These ﬁelds can be derived from the following potentials: i E, ω φ = 0,

A =

(2.87) (2.88)

∇ · A = 0.

(2.89)

Note that physical meaning is attached to the real parts of these complex quantities. As in the previous case, the Hamiltonian with (2.75) has a linear and a quadratic term in A. The quadratic term is related to dynamic polarization and light scattering and (because of its double frequency) to “double quantum” transitions. The linear term in A is more important and gives rise to ﬁrst-order dipole transitions to other states (absorption and emission of radiation). It gives the following term in the Hamiltonian: ˆ dip = iq A · ∇ = − q A · p ˆ. H m m

(2.90)

If the wavelength is large compared to the size of the interacting system, the space dependence of A can be neglected, and A can be considered as a spatially constant vector, although it is still time dependent. Let us consider this term in the Hamiltonian as a perturbation and derive the form of the interaction that will induce transitions between states. In ﬁrst-order perturbation theory, where the wave functions Ψn (r, t) are still solutions ˆ 0 , transitions from state n to state m of the unperturbed Hamiltonian H ˆ 1 matches |En − Em |/h and the occur if the frequency of the perturbation H corresponding matrix element is nonzero: ∞ def ∗ ˆ ˆ ψm (2.91) m|H1 |n = H1 ψn dr = 0. −∞

Thus we need the matrix element m|ˆ p|n, which can be related to the matrix element of the corresponding coordinate: m (2.92) m|ˆ p|n = (Em − En )m|r|n i (the proof is given at the end of this section). The matrix element of the ˆ dip (see (2.90)), summed over all particles, is perturbation H ˆ dip |n = m|H

i (Em − En )A · m| qi r i |n i

= −

(Em − En ) E 0 · μmn , ωmn

(2.93)

36

Quantum mechanics: principles and relativistic eﬀects

where we have made use of (2.87). The term between angular brackets is the transition dipole moment μmn , the matrix element for the dipole moment operator. Note that this dipolar interaction is just an approximation to the total electromagnetic interaction with the ﬁeld. Proof We prove (2.92). We ﬁrst show that m ˆ ˆ = [H p 0 , r], i which follows (for one component) from ˆ 0 , x] = − [H

(2.94)

1 2 [ˆ p , x], 2m

and [p2 , x] = ppx − xpp = ppx − pxp + pxp − xpp = 2p[p, x] = −2ip, because [p, x] = −i. ˆ 0 , r] (for one component): Next we compute the matrix element for [H ˆ 0 x|n − m|xH ˆ 0 |n. ˆ 0 , x]|n = m|H m|[H The last term is simply equal to En m|x|n. The ﬁrst term rewrites by using ˆ 0: the Hermitian property of H ∗ ˆ ∗ ˆ 0∗ ψm ψm H0 (xψn ) dr = (H )xψn dr = Em m|x|n. Collecting terms, (2.92) is obtained. 2.5 Fermions, bosons and the parity rule There is one further basic principle of quantum mechanics that has farreaching consequences for the fate of many-particle systems. It is the rule that particles have a deﬁnite parity. What does this mean? Particles of the same type are in principle indistinguishable. This means that the exchange of two particles of the same type in a many-particle system should not change any observable, and therefore should not change the probability density Ψ∗ Ψ. The wave function itself need not be invariant for particle exchange, because any change of phase exp(iφ) does not change the probability distribution. But if we exchange two particles twice, we return exactly to the original state, so the phase change can only be 0◦ (no change) or 180◦ (change of sign). This means that the parity of the wave function (the change of sign on exchange of two particles) is either positive (even) or negative (odd).

Exercises

37

The parity rule (due to Wolfgang Pauli) says that the parity is a basic, invariant, property of a particle. Thus there are two kinds of particle: fermions with odd parity and bosons with even parity. Fermions are particles with half-integral spin quantum number; bosons have integral spins. Some examples: • fermions (half-integral spin, odd parity): electron, proton, neutron, muon, positron, 3 He nucleus, 3 He atom, D atom; • bosons (integral spin, even parity): deuteron, H-atom, 4 He nucleus, 4 He atom, H2 molecule. The consequences for electrons are drastic! If we have two one-electron orbitals (including spin state) ϕa and ϕb , and we put two non-interacting electrons into these orbitals (one in each), the odd parity prescribes that the total two-particle wave function must have the form ψ(1, 2) ∝ ϕa (1)ϕb (2) − ϕa (2)ϕb (1).

(2.95)

So, if ϕa = ϕb , the wave function cannot exist! Hence, two (non-interacting) electrons (or fermions in general) cannot occupy the same spin-orbital. This is Pauli’s exclusion principle. Note that this exclusion has nothing to do with the energetic (e.g., Coulomb) interaction between the two particles.

Exercises 2.1 2.2 2.3 2.4

Derive (2.30) and (2.31). Show that (2.35) is the Fourier transform of (2.34). See Chapter 12. Show that the width of g ∗ g does not change with time. Show that c2 (dt )2 − (dx )2 = c2 (dt)2 − (dx)2 when dt and dx transform according to the Lorentz transformation of (2.37).

3 From quantum to classical mechanics: when and how

3.1 Introduction In this chapter we shall ask (and possibly answer) the question how quantum mechanics can produce classical mechanics as a limiting case. In what circumstances and for what kind of particles and systems is the classical approximation valid? When is a quantum treatment mandatory? What errors do we make by assuming classical behavior? Are there indications from experiment when quantum eﬀects are important? Can we derive quantum corrections to classical behavior? How can we proceed if quantum mechanics is needed for a speciﬁc part of a system, but not for the remainder? In the following chapters the quantum-dynamical and the mixed quantum/classical methods will be worked out in detail. The essence of quantum mechanics is that particles are represented by a wave function and have a certain width or uncertainty in space, related to an uncertainty in momentum. By a handwaving argument we can already judge whether the quantum character of a particle will play a dominant role or not. Consider a (nearly) classical particle with mass m in an equilibrium system at temperature T , where it will have a Maxwellian velocity distribution (in each direction) with p2 = mkB T . This uncertainty in momentum implies that the particle’s width σx , i.e., the standard deviation of its wave function distribution, will exceed the value prescribed by Heisenberg’s uncertainty principle (see Chapter 2): . σx ≥ √ 2 mkB T

(3.1)

There will be quantum eﬀects if the forces acting on the particle vary appreciably over the width1 of the particle. In condensed phases, with interparticle 1

√ The width we use here is proportional to the de Broglie wavelength Λ = h/ 2πmkB T that ﬁgures in statistical mechanics. Our width is ﬁve times smaller than Λ.

39

40

From quantum to classical mechanics: when and how

Table 3.1 The minimal quantum width in ˚ A of the electron and some atoms at temperatures between 10 and 1000 K, derived from Heisenberg’s uncertainty relation. All values above 0.1 ˚ A are given in bold type

e H D C O I

m(u)

10 K

30 K

100 K

300 K

1000 K

0.000545 1 2 12 16 127

47 1.1 0.78 0.32 0.28 0.098

27 0.64 0.45 0.18 0.16 0.056

15 0.35 0.25 0.10 0.087 0.031

8.6 0.20 0.14 0.058 0.050 0.018

4.7 0.11 0.078 0.032 0.028 0.010

distances of a few ˚ A, this is the case when the width of the particle exceeds, say, 0.1 ˚ A. In Table 3.1 the particle widths are given for the electron and for several atoms for temperatures between 10 and 1000 K. It is clear that electrons are fully quantum-mechanical in all cases (except hot, dilute plasmas with interparticle separations of hundreds of ˚ A). Hydrogen and deuterium atoms are suspect at 300 K but heavier atoms will be largely classical, at least at normal temperatures. It is likely that quantum eﬀects of the heavier atoms can be treated by quantum corrections to a classical model, and one may only hope for this to be true for hydrogen as well. There will be cases where the intermolecular potentials are so steep that even heavy atoms at room temperature show essential quantum eﬀects: this is the case for most of the bond vibrations in molecules. The criterion for classical behavior here is that vibrational frequencies should not exceed kB T /h, which at T = 300 K amounts to about 6 THz, or a wave number of about 200 cm−1 . We may also consider experimental data to judge the importance of quantum eﬀects, at least for systems in thermal equilibrium. In classical mechanics, the excess free energy (excess with respect to the ideal gas value) of a conservative system depends only on the potential energy V (r) and not on the mass of the particles (see Chapter 17): id −N A = A − kB T ln V e−βV (r ) dr. (3.2) Since the ideal gas pressure at a given molar density does not depend on atomic mass either, the phase diagram, melting and boiling points, critical constants, second virial coeﬃcient, compressibility, and several molar properties such as density, heat capacity, etc. do not depend on isotopic composition for a classically behaving substance. Neither do equilibrium

3.1 Introduction

41

Table 3.2 Critical point characteristic for various isotopes of helium, hydrogen and water Tc (K) 4

pc (bar)

Vc (cm3 mol−1 )

He He

5.20 3.34

2.26 1.15

57.76 72.0

H2 HD D2

33.18 35.9 38.3

12.98 14.6 16.3

66.95 62.8 60.3

H2 O D2 O

647.14 643.89

220.64 216.71

56.03 56.28

3

constants as dissociation or association constants or partition coeﬃcients depend on isotopic composition for classical substances. If such properties appear to be dependent on isotopic composition, this is a sure sign of the presence of quantum eﬀects on atomic behavior. Look at a few examples. Table 3.2 lists critical constants for diﬀerent isotopes of helium, hydrogen and water. Table 3.3 lists some equilibrium properties of normal and heavy water. It is not surprising that the properties of helium and hydrogen at (very) low temperatures are strongly isotope dependent. The diﬀerence between H2 O and D2 O is not negligible: D2 O has a higher temperature of maximum density, a higher enthalpy of vaporization and higher molar heat capacity; it appears more “structured” than H2 O. The most likely explanation is that it forms stronger hydrogen bonds as a result of the quantum-mechanical zero-point energy of the intermolecular vibrational and librational modes of hydrogen-bonded molecules. Accurate simulations must either incorporate this quantum behavior, or make appropriate corrections for it. It is instructive to see how the laws of classical mechanics, i.e., Newton’s equations of motion, follow from quantum mechanics. We consider three diﬀerent ways to accomplish this goal. In Section 3.2 we derive equations of motion for the expectation values of position and velocity, following Ehrenfest’s arguments of 1927. A formulation of quantum mechanics which is equivalent to the Schr¨ odinger equation but is more suitable to approach the classical limit, is Feynman’s path integral formulation. We give a short introduction in Section 3.3. Then, in Section 3.4, we consider a formulation of quantum mechanics, originally proposed by Madelung and by de Broglie in 1926/27, and in 1952 revived by Bohm, which represents the evolution

42

From quantum to classical mechanics: when and how

Table 3.3 Various properties of normal and heavy water

◦

melting point ( C) boiling point (◦ C) temperature of maximum density (◦ C) vaporization enthalpy at 3.8 ◦ C (kJ/mol) molar volume at 25 ◦ C (cm3 /mol) molar heat capacity at 25 ◦ C (J K−1 mol−1 ) ionization constant − log[Kw /(mol2 dm−6 )] at 25 ◦ C

H2 O

D2 O

0 100 3.98 44.8 18.07 74.5 13.995

3.82 101.4 11.19 46.5 18.13 83.7 14.951

of the wave function by a ﬂuid of particles which follow trajectories guided by a special quantum force. The application of quantum corrections to equilibrium properties computed with classical methods, and the actual incorporation of quantum eﬀects into simulations, is the subject of following chapters.

3.2 From quantum to classical dynamics In this section we ask the question: Can we derive classical equations of motion for a particle in a given external potential V from the Schr¨ odinger equation? For simplicity we consider the one-dimensional case of a particle of mass m with position x and momentum p = mx. ˙ The classical equations of Newton are p dx = , dt m dV (x) dp = − . dt dx

(3.3) (3.4)

Position and momentum of a quantum particle must be interpreted as the expectation of x and p. The classical force would then be the value of the gradient of V taken at the expectation of x. So we ask whether p dx ? =? , dt m dV dp ? =? − . dt dx x

(3.5) (3.6)

We follow the argument of Ehrenfest (1927). See Chapter 14 for details of the operator formalism and equations of motion. Recall that the expectation A of an observable A over a quantum system

3.2 From quantum to classical dynamics

with wave function Ψ(x, t) is given by ˆ dx, A = Ψ∗ AΨ

43

(3.7)

where Aˆ is the operator of A. From the time-dependent Schr¨ odinger equation the equation of motion (14.64) for the expectation of A follows: i ˆ ˆ dA = [H, A]. dt ˆ A] ˆ is the commutator of H ˆ and A: ˆ Here [H, ˆ A] ˆ =H ˆ Aˆ − AˆH. ˆ [H,

(3.8)

(3.9)

We note that the Hamiltonian is the sum of kinetic and potential energy: 2 ˆ + Vˆ = pˆ + V (x), ˆ =K H (3.10) 2m ˆ but not with Vˆ , while x and that pˆ commutes with K ˆ commutes with Vˆ but ˆ not with K. We shall also need the commutator

. i This follows from inserting the operator for p: [ˆ p, x ˆ] = pˆx ˆ−x ˆpˆ =

(3.11)

∂ψ ∂(xψ) −x = ψ. (3.12) i ∂x i ∂x i Now look at the ﬁrst classical equation of motion (3.5). We ﬁnd using (3.8) that i p dx = [ˆ p2 , x , (3.13) ˆ] = dt 2m m because [ˆ p2 , x ˆ] = pˆpˆx ˆ−x ˆpˆpˆ = pˆpˆx ˆ − pˆx ˆpˆ + pˆx ˆpˆ − x ˆpˆpˆ 2 pˆ. = pˆ[ˆ p, x ˆ] + [ˆ p, x ˆ]ˆ p= i Hence the ﬁrst classical equation of motion is always valid. The second equation of motion (3.6) works out as follows: i ˆ i ˆ dp = [H, pˆ] = [V , pˆ] dt ∂ i ∂ dV i V − V =− . = i ∂x i ∂x dx

(3.14)

(3.15)

This is the expectation of the force over the wave function, not the force

44

From quantum to classical mechanics: when and how

at the expectation of x! When the force is constant, there is no diﬀerence between the two values and the motion is classical, as far as the expectations of x and p are concerned. This is even true if the force depends linearly on x: F (x) = F0 + F (x − x), where F0 = F (x) and F is a constant, because F = F0 + F x − x = F0 . Expanding the force (or potential) in a Taylor series, we see that the leading correction term on the force is proportional to the second derivative of the force times the variance of the wave packet: dV dV 1 d3 V = + (x − x)2 + · · · . (3.16) dx dx x 2! dx3 x The motion is classical if the force (more precisely, the gradient of the force) does not vary much over the quantum width of the particle. This is true even for electrons in macroscopic ﬁelds, as they occur in accelerators and in dilute or hot plasmas; this is the reason that hot plasmas can be treated with classical equations of motion, as long as the electromagnetic interactions are properly incorporated. For electrons near point charges the force varies enormously over the quantum width and the classical approximation fails completely. It is worth mentioning that a harmonic oscillator moves in a potential that has no more than two derivatives, and – as given by the equations derived above – moves according to classical dynamics. Since we know that a quantum oscillator behaves diﬀerently from a classical oscillator (e.g., it has a zero-point energy), this is surprising at ﬁrst sight! But even though the classical equations do apply for the expectation values of position and momentum, a quantum particle is not equal to a classical particle. For example, p2 = p2 . A particle at rest, with p = 0, can still have a kinetic and potential energy. Thus, for classical behavior it is not enough that the expectation of x and p follow classical equations of motion.

3.3 Path integral quantum mechanics 3.3.1 Feynman’s postulate of quantum dynamics While the Schr¨ odinger description of wave functions and their evolution in time is adequate and suﬃcient, the Schr¨ odinger picture does not connect

3.3 Path integral quantum mechanics

45

smoothly to the classical limit. In cases that the particles we are interested in are nearly classical (this will often apply to atoms, but not to electrons) the path integral formulation of quantum mechanics originating from Feynman2 can be more elucidating. This formulation renders a solution to the propagation of wave functions equivalent to that following from Schr¨ odinger’s equation, but has the advantage that the classical limit is more naturally obtained as a limiting case. The method allows us to obtain quantum corrections to classical behavior. In particular, corrections to the classical partition function can be obtained by numerical methods derived from path integral considerations. These path integral Monte Carlo and molecular dynamics methods, PIMC and PIMD, will be treated in more detail in Section 3.3.9. Since the Schr¨ odinger equation is linear in the wave function, the time propagation of the wave function can be expressed in terms of aGreen’s function G(r f , tf ; r 0 , t0 ), which says how much the amplitude of the wave function at an initial position r 0 at time t0 contributes to the amplitude of the wave function at a ﬁnal position r f at a later time tf . All contributions add up to an (interfering) total wave function at time tf : Ψ(r f , tf ) = dr 0 G(r f , tf ; r 0 , t0 )Ψ(r 0 , t0 ). (3.17) The Green’s function is the kernel of the integration. In order to ﬁnd an expression for G, Feynman considers all possible paths {r(t)} that run from position r 0 at time t0 to position r f at time tf . For each path it is possible to compute the mechanical action S as an integral ˙ t) = K − V over that path (see Chapter 15): of the Lagrangian L(r, r, tf ˙ t) dt. S= L(r, r, (3.18) t0

Now deﬁne G as the sum over all possible paths of the function exp(iS/), which represents a phase of the wave function contribution: def G(r f , tf ; r 0 , t0 ) = eiS/. (3.19) all paths

This, of course, is a mathematically dissatisfying deﬁnition, as we do not know how to evaluate “all possible paths” (Fig. 3.1a). Therefore we ﬁrst approximate a path as a contiguous sequence of linear paths over small time steps τ , so a path is deﬁned by straight line segments between the 2

Although Feynman’s ideas date from 1948 (Feynman, 1948) and several articles on the subject are available, the most suitable original text to study the subject is the book by Feynman and Hibbs (1965).

46

From quantum to classical mechanics: when and how

r f (tf )

r(t)

r r r (a)

r 0 (t0 )

tn = tf r r r r ? r r t ti−1 i r Si

(b)

t0

Figure 3.1 (a) Several paths connecting r 0 at time t0 with r f at time tf . One path (thick line) minimizes the action S and represents the path followed by classical mechanics. (b) One path split up into many linear sections (r i−1 , r i ) with actions Si .

initial point r 0 at time t0 , the intermediate points r 1 , . . . , r n−1 at times τ, 2τ, . . . , (n − 1)τ , and the ﬁnal point r n = r f at time tn = t0 + nτ , where n = (tf − t0 )/τ (Fig. 3.1b). Then we construct the sum over all possible paths of this kind by integrating r 1 , . . . , r n−1 over all space. Finally, we take the limit for τ → 0: n G(r f , tf ; r 0 , t0 ) = lim C(τ ) dr 1 · · · dr n−1 exp iSk / , (3.20) τ →0

k=1

where

tk

Sk =

˙ t) dt, L(r, r,

(3.21)

tk−1

over the linear path from r i−1 to r i . Note that a normalizing constant C(τ ) is incorporated which takes care of the normalizing condition for G, assuring that the wave function remains normalized in time. This normalizing constant will depend on τ : the more intermediate points we take, the larger the number of possible paths becomes. Equations (3.20) and (3.21) properly deﬁne the right-hand side (3.19). For small time intervals Sk can be approximated by ˙ t), Sk ≈ τ L(r, r,

(3.22)

with r, r˙ and t evaluated somewhere in – and with best precision precisely halfway – the interval (tk−1 , tk ). The velocity then is r˙ =

r k − r k−1 . τ

(3.23)

3.3 Path integral quantum mechanics

47

3.3.2 Equivalence with the Schr¨ odinger equation Thus far the path integral formulation for the Green’s function has been simply stated as an alternative postulate of quantum mechanics. We must still prove that this postulate leads to the Schr¨ odinger equation for the wave function. Therefore, we must prove that the time derivative of Ψ, as given by the path integral evolution over an inﬁnitesimal time interval, equals −i/ times the Hamiltonian operator acting on the initial wave function. This is indeed the case, as is shown in the following proof for the case of a single particle in cartesian coordinate space in a possibly time-dependent external ﬁeld. The extension to many particles is straightforward, as long as the symmetry properties of the total wave function are ignored, i.e., exchange is neglected. Proof Consider the wave evolution over a small time step τ , from time t to time t + τ : (3.24) Ψ(r, t + τ ) = dr 0 G(r, t + τ ; r 0 , t)Ψ(r 0 , t). Now, for convenience, change to the integration variable δ = r 0 − r and consider the linear path from r 0 to r. The one-particle Lagrangian is given by mδ 2 − V (r, t). (3.25) L= 2τ 2 The action over this path is approximated by mδ 2 − V (r, t)τ, (3.26) 2τ which leads to the following evolution of Ψ: imδ 2 i Ψ(r, t + τ ) ≈ C(τ ) dδ exp exp − V (r, t)τ Ψ(r + δ, t). (3.27) 2τ S≈

We now expand both sides to ﬁrst order in τ , for which we need to expand Ψ(r + δ, t) to second order in δ. The exponent with the potential energy can be replaced by its ﬁrst-order term. We obtain i ∂Ψ ≈ C(τ ) 1 − V τ Ψ + τ ∂t

2 imδ 2 1 ∂2Ψ imδ 2 exp dδ + δx dδ + · · · (y, z) × Ψ exp (3.28) , 2τ 2 ∂x2 2τ where Ψ, its derivatives and V are to be taken at (r, t). The ﬁrst-order term in δ and the second-order terms containing mixed products as δx δy

48

From quantum to classical mechanics: when and how

cancel because they occur in odd functions in the integration. The ﬁrst integral evaluates3 to (ihτ /m)3/2 , which must be equal to the reciprocal of the normalization constant C, as the zeroth-order term in Ψ must leave Ψ unchanged. The second integral evaluates to (iτ /m)(ihτ /m)3/2 and thus the right-hand side of (3.28) becomes 1 2 iτ i Ψ+ ∇ Ψ , 1− Vτ 2 m and the term proportional to τ yields ∂Ψ i 2 2 =− − ∇ Ψ+VΨ , ∂t 2m

(3.29)

which is the Schr¨ odinger equation.

3.3.3 The classical limit From (3.20) we see that diﬀerent paths will in general contribute widely diﬀerent phases, when the total actions diﬀer by a quantity much larger than . So most path contributions will tend to cancel by destructive interference, except for those paths that are near to the path of minimum action Smin . Paths with S − Smin ≈ or smaller add up with roughly the same phase. In the classical approximation, where actions are large compared to , only paths very close to the path of minimum action survive the interference with other paths. So in the classical limit particles will follow the path of minimum action. This justiﬁes the postulate of classical mechanics, that the path of minimum action prescribes the equations of motion (see Chapter 15). Perturbations from classical behavior can be derived by including paths close to, but not coinciding with, the classical trajectory.

3.3.4 Evaluation of the path integral When the Lagrangian can be simply written as ˙ t) = 12 mr˙ 2 − V (r, t), L(r, r,

(3.30)

the action Sk over the short time interval (tk−1 , tk ) can be approximated by Sk = 3

m(r k − r k−1 )2 − V (r k , tk )τ, 2τ

(3.31)

This is valid for one particle in three dimensions; for N particles the 3 in the exponent is replaced by 3N . The use of Planck’s constant h = 2π is not an error!

3.3 Path integral quantum mechanics

and the kernel becomes

dr 1 · · ·

G(r f , tf ; r 0 , t0 ) = lim C(τ ) τ →0

i exp n

k=1

49

dr n−1

m(r k − r k−1 )2 − V (r k , tk )τ 2τ

. (3.32)

The normalization constant C(τ ) can be determined by considering the normalization condition for G. A requirement for every kernel is that it conserves the integrated probability density during time evolution from t1 to t2 : ∗ Ψ (r 2 . t2 )Ψ(r 2 , t2 ) dr 2 = Ψ∗ (r 1 , t1 )Ψ(r 1 , t1 ) dr 1 , (3.33) or, in terms of G: dr 2 dr 1 dr 1 G∗ (r 2 , t2 ; r 1 , t1 )G(r 2 , t2 ; r 1 , t1 )Ψ∗ (r 1 , t1 )Ψ(r 1 , t1 ) (3.34) = dr 1 Ψ∗ (r 1 , t1 )Ψ(r 1 , t1 ) must be valid for any Ψ. This is only true if

dr 2 G∗ (r 2 , t2 ; r 1 , t1 )G(r 2 , t2 ; r 1 , t1 ) = δ(r 1 − r 1 )

(3.35)

for any pair of times t1 , t2 for which t2 > t1 . This is the normalization condition for G. Since the normalization condition must be satisﬁed for any time step, it must also be satisﬁed for an inﬁnitesimal time step τ , for which the path is linear from r 1 to r 2 :

i m(r 2 − r 1 )2 + V (r, t)τ . (3.36) G(r 2 , t + τ ; r 1 , t) = C(τ ) exp 2τ If we apply the normalization condition (3.35) to this G, we ﬁnd that C(τ ) =

ihτ m

−3/2 ,

(3.37)

just as we already found while proving the equivalence with the Schr¨ odinger equation. The 3 in the exponent relates to the dimensionality of the wave function, here taken as three dimensions (one particle in 3D space). For N particles in 3D space the exponent becomes −3N/2.

50

From quantum to classical mechanics: when and how

Proof Consider N particles in 3D space. Now

im ∗ ∗ 2 2 {(r 2 − r 1 ) − (r 2 − r 1 ) } G G dr 2 = C C dr 2 exp 2τ

im 2 im ∗ 2 (r − r 1 ) (r − r 1 ) · r 2 . = C C exp dr 2 exp 2τ 1 τ 1 Using one of the deﬁnitions of the δ-function: +∞ exp(±ikx) dx = 2πδ(k), −∞

or, in 3N dimensions: exp(±ik · r) dr = (2π)3N δ(k), where the delta function of a vector is the product of delta functions of its components, the integral reduces to 3N τ m(r 1 − r 1 ) 3N 3N (2π) δ = (2π) δ(r 1 − r 1 ). τ m Here we have made use of the transformation δ(ax) = (1/a)δ(x). The presence of the delta functions means that the exponential factor before the integral reduces to 1, and we obtain 3N hτ ∗ ∗ δ(r 1 − r 1 ). G G dr 2 = C C m Thus the normalization condition (3.35) is satisﬁed if −3N hτ ∗ . C C= m This is a suﬃcient condition to keep the integrated probability of the wave function invariant in time. But there are many solutions for C, diﬀering by an arbitrary phase factor, as long as the absolute value of C equals the square root of the right-hand side (real) value. However, we do not wish a solution for G that introduces a changing phase into the wave function, and therefore the solution for C found in the derivation of the Schr¨ odinger equation, which leaves not only Ψ∗ Ψ, but also Ψ itself invariant in the limit of small τ , is the appropriate solution. This is (3.37). Considering that we must make n = (tf − t0 )/τ steps to evolve the system

3.3 Path integral quantum mechanics

51

from t0 to tf = tn , we can rewrite (3.32) for a N -particle system as ihτ −3nN/2 dr 1 · · · dr n−1 G(r f , tf ; r 0 , t0 ) = lim τ →0 m n i m(r k − r k−1 )2 − V (r k , tk )τ . (3.38) exp 2τ k=1

Here, r is a 3N -dimensional vector. Note that in the limit τ → 0, the number of steps n tends to inﬁnity, keeping nτ constant. The potential V may still be an explicit function of time, for example due to a time-dependent source. In most cases solutions can only be found by numerical methods. In simple cases with time-independent potentials (free particle, harmonic oscillator) the integrals can be evaluated analytically. In the important case that the quantum system is bound in space and not subjected to a time-dependent external force, the wave function at time t0 can be expanded in an orthonormal set of eigenfunctions φn of the Hamiltonian: Ψ(r, t0 ) = an φn (r). (3.39) n

As the eigenfunction φn develops in time proportional to exp(−iEn t/), we know the time dependence of the wave function: i an φn (r) exp − En (t − t0 ) . (3.40) Ψ(r, t) = n ¿From this it follows that the kernel must have the following form: G(r, t; r 0 , t0 ) = n φn (r)φ∗n (r 0 ) exp − i En (t − t0 ) . (3.41) This is easily seen by applying G to the initial wave function m am φm (r 0 ) and integrating over r 0 . So in this case the path integral kernel can be expressed in the eigenfunctions of the Hamiltonian. 3.3.5 Evolution in imaginary time A very interesting and useful connection can be made between path integrals and the canonical partition function of statistical mechanics. This connection suggests a numerical method for computing the thermodynamic properties of systems of quantum particles where the symmetry properties of wave functions and, therefore, eﬀects of exchange can be neglected. This is usually the case when systems of atoms or molecules are considered at normal temperatures: the repulsion between atoms is such that the

52

From quantum to classical mechanics: when and how

quantum-mechanical exchange between particles (nuclei) is irrelevant. The quantum eﬀects due to symmetry properties (distinguishing fermions and bosons) are completely drowned in the quantum eﬀects due to the curvature of the interatomic potentials within the de Broglie wavelength of the nuclei. Consider the quantum-mechanical canonical partition function of an N particle system exp(−βEn ), (3.42) Q= n

where the sum is to be taken over all quantum states (not energy levels!) and β equals 1/kB T . Via the free energy relation (A is the Helmholtz free energy) A = −kB T ln Q

(3.43)

and its derivatives with respect to temperature and volume, all relevant thermodynamic properties can be obtained. Unfortunately, with very few exceptions under idealized conditions, we cannot enumerate all quantum states and energy levels of a complex system, as this would mean the determination of all eigenvalues of the full Hamiltonian of the system. Since the eigenfunctions φn of the system form an orthonormal set, we can also write (3.42) as φn (r)φ∗n (r) exp(−βEn ). (3.44) Q = dr n

Now compare this with the expression for the path integral kernel of (3.41). Apart from the fact that initial and ﬁnal point are the same (r), and the form is integrated over dr, we see that instead of time we now have −iβ. So the canonical partition function is closely related to a path integral over negative imaginary time. The exact relation is Q = drG(r, −iβ; r, 0), (3.45) with (inserting τ = −iβ/n into (3.38)) G(r, −iβ; r, 0) = lim C(n) dr 1 · · · dr n−1 n→∞

n nm 1 exp −β (r k − r k−1 )2 + V (r k ) , (3.46) 22 β 2 n k=1

3.3 Path integral quantum mechanics

53

q COC

q q Q C q QQ sq q C J J q q q ^ J * q

1qH q I @ H H

@q q j

H q /r n−1 BM B @ q ) CO Bq q @ i P C Rr r = r 0 = r n @

PP C r1 C

q r2 Figure 3.2 A closed path in real space and imaginary time, for the calculation of quantum partition functions.

where

C(n) =

h2 β 2πnm

−3nN/2 ,

(3.47)

and r 0 = r n = r. Note that r stands for a 3N -dimensional cartesian vector of all particles in the system. Also note that all paths in the path integral are closed: they end in the same point where they start (Fig. 3.2). In the multiparticle case, the imaginary time step is made for all particles simultaneously; each particle therefore traces out a three-dimensional path. A path integral over imaginary time does not add up phases of diﬀerent paths, but adds up real exponential functions over diﬀerent paths. Only paths with reasonably-sized exponentials contribute; paths with highly negative exponents give negligible contributions. Although it is diﬃcult to imagine what imaginary-time paths mean, the equations derived for realtime paths can still be used and lead to real integrals.

3.3.6 Classical and nearly classical approximations Can we easily see what the classical limit is for imaginary-time paths? Assume that each path (which is closed anyway) does not extend very far from its initial and ﬁnal point r. Assume also that the potential does not vary

54

From quantum to classical mechanics: when and how

appreciably over the extent of each path, so that it can be taken equal to V (r) for the whole path. Then we can write, instead of (3.46): G(r, −iβ; r, 0) = exp(−βV (r)) lim C(n) dr 1 · · · dr n−1 n→∞ n nm 2 (3.48) exp − 2 (r k − r k−1 ) . 2 β k=1

The expression under the limit sign yields (2πmkB T /h2 )3N/2 , independent of the number of nodes n. The evaluation of the multiple integral is not entirely trivial, and the proof is given below. Thus, after integrating over r we ﬁnd the classical partition function 2πmkB T 3N/2 (3.49) e−βV (r ) dr. Q= h2 Since the expression is independent of n, there is no need to take the limit for n → ∞. Therefore the imaginary-time path integral without any intervening nodes also represents the classical limit. Note that the integral is not divided by N !, since the indistinguishability of the particles has not been introduced in the path integral formalism. Therefore we cannot expect that path integrals for multiparticle systems will treat exchange eﬀects correctly. For the application to nuclei in condensed matter, which are always subjected to strong short-range repulsion, exchange eﬀects play no role at all. Proof We prove that n nm 2πmkB T 3N/2 2 (r k − r k−1 ) . = C(n) dr 1 . . . dr n−1 exp − 2 2 β h2 k=1

First make a coordinate transformation from r k to sk = r k − r 0 , k = 1, . . . , n − 1. Since the Jacobian of this transformation equals one, dr can be replaced by ds. Inspection of the sum shows that the integral I now becomes I = ds1 · · · dsn−1 exp[−α{s21 +(s2 −s1 )2 +· · ·+(sn−1 −sn−2 )2 +s2n−1 }], where α=

nm . 22 β

The expression between { } in s can be written in matrix notation as sT An s,

3.3 Path integral quantum mechanics

55

with An a symmetric tridiagonal (n − 1) × (n − 1) matrix with 2 along the diagonal, −1 along both subdiagonals, and zero elsewhere. The integrand becomes a product of independent Gaussians after an orthogonal transformation that diagonalizes An ; thus the integral depends only on the product of eigenvalues, and evaluates to π 3(n−1)N/2 I= (det An )−3N/2 . α The determinant can be easily evaluated from its recurrence relation, det An = 2 det An−1 − det An−2 , and turns out to be equal to n. Collecting all terms, and replacing C(n) by (3.47), we ﬁnd the required result. Note that the number of nodes n cancels: the end result is valid for any n. In special cases, notably a free particle and a particle in an isotropic harmonic potential, analytical solutions to the partition function exist. When the potential is not taken constant, but approximated by a Taylor expansion, quantum corrections to classical simulations can be derived as perturbations to a properly chosen analytical solution. These applications will be treated in Section 3.5; here we shall derive the analytical solutions for the two special cases mentioned above.

3.3.7 The free particle The canonical partition function of a system of N free (non-interacting) particles is a product of 3N independent terms and is given by Q = lim Q(n) , (n)

Q

q (n)

n→∞ (n) 3N

= (q ) , 2πmn n/2 dx0 · · · dxn−1 = h2 β

n exp −a (xk − xk−1 )2 ,

(3.50)

k=1

a =

nm ; xn = x0 . 22 β

(3.51)

The sum in the exponent can be written in matrix notation as n k=1

(xk − xk−1 )2 = xT Ax = yT Λy,

(3.52)

56

From quantum to classical mechanics: when and how

where A is a symmetric cyclic tridiagonal matrix: ⎛ ⎞ 2 −1 0 −1 ⎜ −1 2 −1 0 ⎟ ⎜ ⎟ ⎜ ⎟ A=⎜ ... ⎟ ⎝ 0 −1 2 −1 ⎠ −1 0 −1 2

(3.53)

and y is a set of coordinates obtained by the orthogonal transformation T of x that diagonalizes A to the diagonal matrix of eigenvalues Λ = diag (λ0 , . . . , λn−1 ): y = Tx, T−1 = TT , xT Ax = yT TATT y = yT Λy.

(3.54)

There is one zero eigenvalue, corresponding to an eigenvector proportional to the sum of xk , to which the exponent is invariant. This eigenvector, which we shall label “0,” must be separated. The eigenvector y0 is related to the centroid or average coordinate xc : 1 xk , n n−1

def

xc =

(3.55)

k=0

√ 1 y0 = √ (1, 1, . . . , 1)T = rc n. n

(3.56)

Since the transformation is orthogonal, its Jacobian equals 1 and integration over dx can be replaced by integration over dy. Thus we obtain

n−1 n/2 2πmn q (n) = n1/2 dxc dy1 · · · dyn−1 exp −a λk yk2 . h2 β k=1 (3.57) Thus the distribution of node coordinates (with respect to the centroid) is a multivariate Gaussian distribution. Its integral equals

n−1 π (n−1)/2 −1/2 Πn−1 λk yk2 = λk . (3.58) dy1 · · · dyn−1 exp −a k=1 a k=1

The product of non-zero eigenvalues of matrix A turns out to be equal to n2 (valid for any n). Collecting terms we ﬁnd that the partition function equals the classical partition function for a 1D free particle for any n: 2πm 1/2 (n) (3.59) q = dxc , h2 β as was already shown to be the case in (3.49).

3.3 Path integral quantum mechanics

57

Table 3.4 Intrinsic variance of a discrete imaginary-time path for a one-dimensional free particle as a function of the number of nodes in the path, in units of 2 β/m n

σ2

2 3 4 5 6 7

0.062 500 0.074 074 0.078 125 0.080 000 0.081 019 0.081 633

σ2

n 8 9 10 20 30 40

0.082 031 0.082 305 0.082 500 0.083 125 0.083 241 0.083 281

n 50 60 70 80 90 ∞

σ2 0.083 300 0.083 310 0.083 316 0.083 320 0.083 323 0.083 333

The variance σ 2 of the multivariate distribution is given by

1 (xk − xc )2 n

1 2 1 −1 2 β −1 = yk = λk = 2 λk . σ = n 2an n m k=0 k=1 k=1 k=1 (3.60) We shall call this the intrinsic variance in order to distinguish this from the distribution of the centroid itself. The sum of the inverse non-zero eigenvalues has a limit of n2 /12 = 0.083 33 . . . n2 for n → ∞. In Table 3.4 the intrinsic variance of the node distribution is given for several values of n, showing that the variance quickly converges: already 96% of the limiting variance is realized by a path consisting of ﬁve nodes. 2 def

n−1

n−1

n−1

n−1

3.3.8 Non-interacting particles in a harmonic potential The other solvable case is a system of N non-interacting particles that each reside in an external harmonic potential V = 12 mω 2 r 2 . Again, the partition function is the product of 3N 1D terms: Q = lim (q (n) )3N , n→∞ 2πmn n/2 (n) dx0 · · · dxn−1 = q h2 β

n n exp −a (xk − xk−1 )2 − b x2k , k=1

b =

βmω 2 , 2n

(3.61)

k=1

(3.62)

58

From quantum to classical mechanics: when and how

a is deﬁned in (3.51). We can separate the centroid harmonic term as follows: 1 1 2 1 2 xk = x2c + (xk − xc )2 = x2c + yk , n n n n

k=1

n

k=1

n−1

(3.63)

k=1

and obtain for the 1D partition function 1 2πmn n/2 1/2 (n) exp[− βmω 2 x2c ] dxc q = n 2 h β 2 × dy1 · · · dyn−1 exp −ayT By ,

(3.64)

where B is a modiﬁed version of A from (3.53): (βω)2 1. (3.65) n2 The eigenvalues of B are equal to the eigenvalues of A increased with (βω/n)2 and the end result is

2πmn n/2 1/2 1 (n) 2 2 = n q exp − βmω xc dxc h2 β 2 × Πn−1 (3.66) exp[−aλk yk2 ] dyk k=1 B=A+

1 n−1 −1/2 1 2πm 1/2 ∞ 2 2 exp − βmω xc dxc Π λ , = h2 β 2 n2 k=1 k −∞ (βω)2 λk = λk + . (3.67) n2

So we need the product of all eigenvalues λ1 , . . . , λn−1 . For n → ∞ the partition function goes to the quantum partition function of the harmonic oscillator (see (17.84) on page 472), which can be written in the form

2πm 1/2 1 (∞) 2 2 q = exp − βmω xc dxc h2 β 2 ξ , ξ = βω. (3.68) × 2 sinh(ξ/2) The last term is an exact quantum correction to the classical partition function. From this it follows that ξ 1 n−1 lim , (3.69) Π λ = n→∞ n2 k=1 k 2 sinh(ξ/2) which can be veriﬁed by direct evaluation of the product of eigenvalues for large n. Figure 3.3 shows the temperature dependence of the free energy and

3.3 Path integral quantum mechanics

59

Helmholtz free energy A/hν 0.6 0.4

∞ 50 10 5 3

2 1

0.2

0.2

0.4

0.6

0.8

1

temperature kT/hν

internal energy U/hν 1 0.8 0.6

∞ 50

0.4 0.2

10 5

3

2 1

0.2

0.4

0.6

0.8 1 temperature kT/hν

Figure 3.3 Free energy A and energy U of a 1D harmonic oscillator evaluated as an imaginary-time path integral approximated by n nodes. The curves are labeled with n; the broken line represents the classical approximation (n = 1); n = ∞ represents the exact quantum solution. Energies are expressed in units of hν = ω.

the energy of a 1D harmonic oscillator, evaluated by numerical solution of (3.67) for several values of n. The approximation fairly rapidly converges to the exact limit, but for low temperatures a large number of nodes is needed, while the limit for T = 0 (A = U = 0.5ω) is never reached correctly. The values of U , as plotted in Fig. 3.3, were obtained by numerical differentiation as U = A − T dA/dT . One can also obtain U = −∂ ln q/∂β by diﬀerentiating q of (3.67), yielding U (n) =

n−1 1 1 1 . + β2 ω 2 2 β n λk k=1

(3.70)

60

From quantum to classical mechanics: when and how

The ﬁrst term is the internal energy of the classical oscillator while the second term is a correction caused by the distribution of nodes relative to the centroid. Both terms consist of two equal halves representing kinetic and potential energy, respectively. The intrinsic variance of the distribution of node coordinates, i.e., relative to the centroid, is – as in (3.60) – given by 2 = σintr

n−1 2 β 1 1 . m n2 λk

(3.71)

k=1

We immediately recognize the second term of (3.70). If we add the “classical” variance of the centroid itself 2 = σcentroid

1 , βmω 2

(3.72)

we obtain the total variance, which can be related to the total energy: 2 2 + σintr = σ 2 = σcentroid

U . mω 2

(3.73)

This is compatible with U being twice the potential energy Upot , which equals 0.5mω 2 x2 . Its value as a function of temperature is proportional to the curve for U (case n → ∞) in Fig. 3.3. As is to be expected, the total variance for n → ∞ equals the variance of the wave function, averaged over the occupation of quantum states v: x2 =

∞ v=0

P (v)x2 v =

∞ P (v)Ev v=0

mω 2

=

U qu , mω 2

(3.74)

where P (v) is the probability of occurrence of quantum state v, because the variance of the wave function in state v is given by (v + 12 ) Ev = . (3.75) x2 v = Ψ∗v x2 Ψ dx = mω mω 2 Note that this relation between variance and energy is not only valid for a canonical distribution, but for any distribution of occupancies. We may summarize the results for n → ∞ as follows: a particle can be considered as a distribution of imaginary-time closed paths around the centroid of the particle. The intrinsic (i.e., with respect to the centroid) spatial distribution for a free particle is a multivariate Gaussian with a variance of 2 β/(12m) in each dimension. The variance (in each dimension) of the distribution for a particle in an isotropic harmonic well (with force

3.3 Path integral quantum mechanics

61

intrinsic variance σ2 (units: h/mω) 1 0.8 free particle σ2 =

h2 12 m kT

0.6 0.4 0.2

harmonic oscillator 0.2

0.4

0.6

0.8 1 temperature kT/hω

Figure 3.4 The intrinsic variance in one dimension of the quantum imaginary-time path distribution for the free particle and for the harmonic oscillator.

constant mω 2 ) is given by 2 σintr

U qu − U cl = = 2 mω mω

1 ξ 1 coth − 2 2 ξ

, ξ = βω.

(3.76)

For high temperatures (small ξ) this expression goes to the free-particle value 2 β/12m; for lower temperatures the variance is reduced because of the quadratic potential that restrains the spreading of the paths; the lowtemperature (ground state) limit is /(2mω). Figure 3.4 shows the intrinsic variance as a function of temperature.

3.3.9 Path integral Monte Carlo and molecular dynamics simulation The possibility to use imaginary-time path integrals for equilibrium quantum simulations was recognized as early as 1962 (Fosdick, 1962) and developed in the early eighties (Chandler and Wolynes, 1981; Ceperley and Kalos, 1981; and others). See also the review by Berne and Thirumalai (1986). Applications include liquid neon (Thirumalai et al., 1984), hydrogen diﬀusion in metals (Gillan, 1988), electrons in fused salts (Parrinello and Rahman, 1984, using a molecular dynamics variant), hydrogen atoms and muonium in water (de Raedt et al., 1984), and liquid water (Kuharski and Rossky, 1985; Wallqvist and Berne, 1985).

62

From quantum to classical mechanics: when and how

If the potential is not approximated, but evaluated for every section of the path, the expression for Q becomes Q= ×

2πmkB T h2

dr 1 · · ·

3N/2

dr lim

n→∞

2πmkB T h2

3(n−1)N/2 n3nN/2

n nm V (r ) k , (3.77) dr n−1 exp −β (r k − r k−1 )2 + 22 β 2 n

k=1

with r = r n = r 0 . The constant after the limit symbol exactly equals the inverse of the integral over the harmonic terms only, as was shown in the proof of the classical limit given on page 54:

dr 1 · · ·

2πmkB T h2

−3(n−1)N/2

dr n−1 exp −β

n−1 k=1

n−3nN/2 =

nm (r k − r k−1 )2 22 β 2

(3.78)

.

(3.79)

It therefore “compensates” in the partition function for the harmonic terms in the extra degrees of freedom that are introduced by the beads. Interestingly, the expression for Q in (3.77) is proportional to the partition function of a system of particles, where each particle i is represented by a closed string of beads (a “necklace”), with two adjacent beads connected by a harmonic spring with spring constant κi =

nmi , 2 β 2

(3.80)

and feeling 1/n of the interparticle potential at the position of each bead. The interaction V (r i − r j ) between two particles i and j acts at the k-th step between the particles positioned at the k-th node r k . Thus the k-th node of particle i interacts only with the k-th node of particle j (Fig. 3.5), with a strength of 1/n times the full interparticle interaction. The propagator (3.48) used to derive the “string-of-beads” homomorphism, is a high-temperature free particle propagator, which – although in principle exact – converges slowly for bound particles in potential wells at low temperature. Mak and Andersen (1990) have devised a “low-temperature” propagator that is appropriate for particles in harmonic wells. It contains the resonance frequency (for example derived from the second derivative of the potential) and converges faster for bound states with similar frequencies.

3.3 Path integral quantum mechanics 16 COC 14 8Q 17 s Q C 9 J 15 ^ J 18 13 *7 6 111

H 10 I @ j H @5 H 12

/ BM 3 19 @ B4 ) CO 1P R0 @ C iP C2

63

16 Qs 15 Q 17 BM B 18 14 19 11P 13 iP k Q + 10 0 Q12 8 ) 4 9 12

3 1 AU 3 j7 H 36H 5

Figure 3.5 The paths of two interacting particles. Interactions act between equallynumbered nodes, with strength 1/n.

For this propagator the Boltzmann term in (3.77) reads

n−1 mω(r k − r k−1 )2 2{cosh(βω/n) − 1} + V (r k ) . (3.81) exp −β 2β sinh(βω/n) βω sinh(βω/n) k=1

The system consisting of strings of beads can be simulated in equilibrium by conventional classical Monte Carlo (MC) or molecular dynamics (MD) methods. If MD is used, and the mass of each particle is evenly distributed over its beads, the time step will become quite small. The oscillation frequency of the bead harmonic oscillators is approximately given by nkB T /h, which amounts to about 60 THz for a conservative number of ten beads per necklace, at T = 300 K. Taking 50 steps per oscillation period then requires a time step as small as 0.3 fs. Such PIMC or PIMD simulations will yield a set of necklace conﬁgurations (one necklace per atom) that is representative for an equilibrium ensemble at the chosen temperature. The solution is in principle exact in the limit of an inﬁnite number of beads per particle, if exchange eﬀects can be ignored. While the PIMC and PIMD simulations are valid for equilibrium systems, their use in non-equilibrium dynamic simulations is questionable. One may equilibrate the “quantum part,” i.e., the necklace conﬁgurations, at any given conﬁguration of the geometric centers of the necklaces, either by MC or MD, and compute the necklace-averaged forces between the particles. Then one may move the system one MD step ahead with those eﬀective forces. In this way a kind of quantum dynamics is produced, with the momentum change given by quantum-averaged forces, rather than by forces evaluated at the quantum-averaged positions. This is exactly what should be done for the momentum expectation value, according to the derivation by Ehrenfest (see page 43). One may hope that this method of computing

64

From quantum to classical mechanics: when and how

forces incorporates essential quantum eﬀects in the dynamics of the system. However, this cannot be proven, as the wave function distribution that is generated contains no memory of the past dynamics as it should in a full quantum-dynamical treatment. Neither does this kind of “quantum dynamics” produce a bifurcation into more than one quantum state. Note that this method cannot handle exchange.4 In Section 3.5 we’ll return to the path-integral methods and employ them to make approximate quantum corrections to molecular dynamics. 3.4 Quantum hydrodynamics In this section a diﬀerent approach to quantum mechanics is considered, which originates from Madelung (1926) and de Broglie (1927) in the 1920’s, and was revived by Bohm (1952a, 1952b) in the ﬁfties. It survived the following decades only in the periphery of the main stream of theoretical physics, but has more recently regained interest because of its applicability to simulation. The approach is characterized by the use of a classical model consisting of a system of particles or a classical ﬂuid that – under certain modiﬁed equations of motions that include a quantum force – behaves according to the Schr¨ odinger equation. Models based on deterministic ﬂuid dynamics are known as quantum hydrodynamics or Madelung ﬂuid. If the ﬂuid is represented by a statistical distribution of point particles, the term Bohmian particle dynamics is often used. Particles constituting the quantum ﬂuid have also been called “beables,” in analogy and contrast to the “observables” of traditional quantum mechanics (Bell 1976, 1982; Vink 1993). The quantum force is supposed to originate from a wave that accompanies the quantum particles.5 In addition to the interpretation in terms of a ﬂuid or a distribution of particles behaving according to causal relations, several attempts have been made to eliminate the quantum force and ascribe its eﬀects to the diﬀusional behavior of particles that undergo stochastic forces due to some unknown external agent. Such quantum stochastic dynamics methods (F´enyes 1952; Weizel 1954; Kershaw 1964; Nelson 1966; Guerra 1981) will not be considered further in our context as they have not yet led to useful simulation techniques. It is possible to rewrite the time-dependent Schr¨ odinger equation in a diﬀerent form, such that the square of the wave function can be interpreted 4 5

Exchange can be introduced into path integral methods, see Roy and Voth (1999), and should never be applied to electrons in systems with more than one electron. See for an extensive description of the particle interpretation, including a discussion of its origins, the book of Holland (1993).

3.4 Quantum hydrodynamics

65

as the density of a frictionless classical ﬂuid evolving under hydrodynamic equations, with the ﬂuid particles subjected to a quantum-modiﬁed force. The force consists of two parts: the potential force, which is minus the gradient of the potential V , and a quantum force, which is minus the gradient of a quantum potential Q. The latter is related to the local curvature of the density distribution. This mathematical equivalence can be employed to generate algorithms for the simulation of the evolution of wave packets, but it can also be used to evoke a new interpretation of quantum mechanics in terms of hidden variables (positions and velocities of the ﬂuid particles). Unfortunately, the hidden-variable aspect has dominated the literature since Bohm. Unfortunately, because any invocation of hidden variables in quantum mechanics is in conﬂict with the (usual) Copenhagen interpretation of quantum mechanics, and rejected by the main stream physicists. The Copenhagen interpretation6 considers the wave function of a system of particles as no more than an expression from which the probability of the outcome of a measurement of an observable can be derived; it attaches no meaning to the wave function as an actual, physically real, attribute of the system. The wave function expresses all there is to know about the system from the point of view of an external observer. Any interpretation in terms of more details or hidden variables does not add any knowledge that can be subjected to experimental veriﬁcation and is therefore considered by most physicists as irrelevant. We shall not enter the philosophical discussion on the interpretation of quantum mechanics at all, as our purpose is to simulate quantum systems including the evolution of wave functions. But this does not prevent us from considering hypothetical systems of particles that evolve under speciﬁed equations of motion, when the wave function evolution can be derived from the behavior of such systems by mathematical equivalence. A similar equivalence is the path-integral Monte Carlo method to compute the evolution of ensemble-averaged quantum behavior (see Section 3.3.9), where a ring of particles interconnected by springs has a classical statistical behavior that is mathematically equivalent to the ensemble-averaged wave function evolution of a quantum particle. Of course, such equivalences are only useful when they lead to simulation methods that are either simpler or more eﬃcient than the currently available ones. One of the reasons that interpretations of the quantum behavior in terms of distributions of particles can be quite useful in simulations is that such interpretations allow the construction of quantum trajectories which can be more naturally combined with 6

Two articles, by Heisenberg (1927) and Bohr (1928), have been reprinted, together with a comment on the Copenhagen interpretation by A. Herrmann, in Heisenberg and Bohr (1963).

66

From quantum to classical mechanics: when and how

classical trajectories. They may oﬀer solutions to the problem how to treat the back reaction of the quantum subsystem to the classical degrees of freedom. The ontological question of existence of the particles is irrelevant in our context and will not be considered.

3.4.1 The hydrodynamics approach Before considering a quantum particle, we shall ﬁrst summarize the equations that describe the time evolution of a ﬂuid. This topic will be treated in detail in Chapter 9, but we only need a bare minimum for our present purpose. Assume we have a ﬂuid with mass density mρ(r, t) and ﬂuid velocity u(r, t). We shall consider a ﬂuid conﬁned within a region of space such that ρ dr = 1 at all times, so the total mass of the ﬂuid is m. The ﬂuid is homogeneous and could consist of a large number N → ∞ of “particles” with mass m/N and number density N ρ, but ρ could also be interpreted as the probability density of a single particle with mass m. The velocity u(r, t) then is the average velocity of the particle, averaged over the distribution ρ(r, t), which in macroscopic ﬂuids is often called the drift velocity of the particle. It does not exclude that the particle actual velocity has an additional random contribution. We deﬁne the ﬂux density J as J = ρu.

(3.82)

Now the fact that particles are not created or destroyed when they move, or that total density is preserved, implies the continuity equation ∂ρ + ∇ · J = 0. (3.83) ∂t This says that the outward ﬂow J ·dS over the surface of a (small) volume V , which equals the integral of the divergence ∇ · J of J over that volume, goes at the expense of the integrated density present in that volume. There is one additional equation, expressing the acceleration of the ﬂuid by forces acting on it. This is the equation of motion. The local acceleration, measured in a coordinate system that moves with the ﬂow, and indicated by the material or Lagrangian derivative D/Dt, is given by the force f acting per unit volume, divided by the local mass per unit volume Du f (r) = . Dt mρ

(3.84)

3.4 Quantum hydrodynamics

67

The material derivative of any attribute A of the ﬂuid is deﬁned by ∂A DA def ∂A ∂A dx ∂A dy ∂A dz + + + = + u · ∇A = Dt ∂t ∂x dt ∂y dt ∂z dt ∂t

(3.85)

and thus (3.84) can be written as the Lagrangian equation of motion ∂u Du = mρ + (u · ∇)u = f (r). (3.86) mρ Dt ∂t The force per unit volume consists of an external component f ext = −ρ∇V (r, t)

(3.87)

due to an external potential V (r, t), and an internal component f int = ∇ · σ,

(3.88)

where σ is the local stress tensor, which for isotropic frictionless ﬂuids is diagonal and equal to minus the pressure (see Chapter 9 for details). This is all we need for the present purpose. Let us now return to a system of quantum particles.7 For simplicity we consider a single particle with wave function Ψ(r, t) evolving under the timedependent Schr¨ odinger equation (5.22). Generalization to the many-particle case is a straightforward extension that we shall consider later. Write the wave function in polar form: √

Ψ(r, t) = R(r, t) exp[iS(r, t)/].

(3.89)

Here, R = Ψ∗ Ψ and S are real functions of space and time. Note that R is non-negative and that S will be periodic with a period of 2π, but S is undeﬁned when R = 0. In fact, S can be discontinuous in nodal planes where R = 0; for example, for a real wave function S jumps discontinuously from 0 to π at a nodal plane. The Schr¨ odinger equation, ∂Ψ 2 2 =− ∇ Ψ + V (r, t)Ψ, (3.90) ∂t 2m can be split into an equation for the real and one for the imaginary part. We then straightforwardly obtain two real equations for the time dependence of R and S, both only valid when R = 0: i

∂R 1 R 2 = − ∇R · ∇S − ∇ S, ∂t m 2m 1 ∂S = − (∇S)2 − (V + Q), ∂t 2m 7

(3.91) (3.92)

We follow in essence the lucid treatment of Madelung (1926), with further interpretations by Bohm (1952a, 1952b), and details by Holland (1993).

68

From quantum to classical mechanics: when and how

where Q is deﬁned as def

Q(r) = −

2 ∇2 R . 2m R

(3.93)

The crucial step now is to identify ∇S/m as the local ﬂuid velocity u: def

u(r, t) =

∇S . m

(3.94)

This is entirely reasonable, since the expectation value of the velocity equals the average of ∇S/m over the distribution ρ: 2 2 ∇S v = k = dr. Ψ∗ ∇Ψ dr = R(∇R) dr + R2 m im im m The ﬁrst term is zero because R vanishes over the integration boundaries, so that ∇S dr. (3.95) v = R2 m Applying this identiﬁcation to (3.91), and writing ρ(r, t) = R2 ,

(3.96)

∂ρ + ∇ · (ρu) = 0. ∂t

(3.97)

we ﬁnd that

This is a continuity equation for ρ (see (3.83))! Equation (3.92) becomes m

∂u = −∇( 12 mu2 + V + Q). ∂t

(3.98)

The gradient of u2 can be rewritten as 1 2 2 ∇(u )

= (u · ∇)u,

as will be shown below; therefore ∂u Du m + (u · ∇)u = m = −∇(V + Q), ∂t Dt

(3.99)

(3.100)

which is the Lagrangian equation of motion, similar to (3.86). The local force per unit volume equals −ρ∇(V + Q). This force depends on position, but not on velocities, and thus the ﬂuid motion is frictionless, with an external force due to the potential V and an “internal force” due to the quantum potential Q.

3.4 Quantum hydrodynamics

69

Proof We prove (3.99). Since u is the gradient of S, the u-ﬁeld is irrotational: curl u = 0, for all regions of space where ρ = 0. Consider the x-component of the gradient of u2 : ∂uy 1 ∂ux ∂uz ∂ux ∂ux ∂ux (∇u2 )x = ux + uy + uz = ux + uy + uz 2 ∂x ∂x ∂x ∂x ∂y ∂z (3.101) = (u · ∇)ux , because curl u = 0 implies that ∂uy ∂ux ∂uz ∂ux = and = . ∂x ∂y ∂x ∂z

The quantum potential Q, deﬁned by (3.93), can also be expressed in derivatives of ln ρ: 2 2 (3.102) ∇ ln ρ + 12 (∇ ln ρ)2 , 4m which may be more convenient for some applications. The quantum potential is some kind of internal potential, related to the density distribution, as in real ﬂuids. One may wonder if a simple deﬁnition for the stress tensor (3.88) exists. It is indeed possible to deﬁne such a tensor (Takabayasi, 1952), for which Q=−

f int = −ρ∇Q = ∇σ,

(3.103)

when we deﬁne def

σ =

2 ρ∇∇ ln ρ. 4m

(3.104)

This equation is to be read in cartesian coordinates (indexed by α, β, . . .) as σαβ =

2 ∂ 2 ln ρ ρ . 4m ∂xα ∂xβ

(3.105)

3.4.2 The classical limit In the absence of the quantum force Q the ﬂuid behaves entirely classically; each ﬂuid element moves according to the classical laws in a potential ﬁeld V (r), without any interaction with neighboring ﬂuid elements belonging to the same particle. If the ﬂuid is interpreted as a probability density, and the initial distribution is a delta-function, representing a point particle, then in the absence of Q the distribution will remain a delta function and follow a classical path. Only under the inﬂuence of the quantum force will the

70

From quantum to classical mechanics: when and how

distribution change with time. So the classical limit is obtained when the quantum force (which is proportional to 2 ) is negligible compared to the interaction force. Note, however, that the quantum force will never be small for a point particle, and even near the classical limit particles will have a non-zero quantum width.

3.5 Quantum corrections to classical behavior For molecular systems at normal temperatures that do not contain very fast motions of light particles, and in which electronically excited states play no role, classical simulations will usually suﬃce to obtain relevant dynamic and thermodynamic behavior. Such simulations are in the realm of molecular dynamics (MD),which is the subject of Chapter 6. Still, when high-frequency motions do occur or when lighter particles and lower temperatures are involved and the quantum wavelength of the particles is not quite negligible compared to the spatial changes of interatomic potentials, it is useful to introduce quantum eﬀects as a perturbation to the classical limit and evaluate the ﬁrst-order quantum corrections to classical quantities. This can be done most simply as a posterior correction to quantities computed from unmodiﬁed classical simulations, but it can be done more accurately by modifying the equations of motions to include quantum corrections. In general we shall be interested to preserve equilibrium and long-term dynamical properties of the real system by the classical approximation. This means that correctness of thermodynamic properties has priority over correctness of dynamic properties. As a starting point we may either take the quantum corrections to the partition function, as described in Chapter 17, Section 17.6 on page 472, or the imaginary-time path-integral approach, where each particle is replaced by a closed string of n harmonically interacting beads (see Section 3.3 on page 44). The latter produces the correct quantum partition function. In the next section we shall start with the Feynman–Hibbs quantum-corrected pair potential and show that this potential results in corrections to thermodynamic quantities that agree with the quantum corrections known from statistical mechanics.

3.5.1 Feynman-Hibbs potential In Sections 3.3.7 (page 55) and 3.3.8 (page 57) the intrinsic quantum widths of free and of harmonically-bound particles were derived. Both are Gaussian

3.5 Quantum corrections to classical behavior

71

distributions, with variances: 2 (free particle) 12 mkB T

ω kT 1 coth − (harmonic particle) σ2 = mω 2 2kB T ω

σ2 =

(3.106) (3.107)

These can be used to modify pair potentials. We shall avoid the complications caused by the use of a reference potential,8 needed when the harmonic width is used, and only use the free particle distribution. Feynman and Hibbs (1965) argued that each pair interaction Vij (rij ) = U (r) between two particles i and j with masses mi and mj should be modiﬁed by a 3D convolution with the free-particle intrinsic quantum distribution:

s2 FH 2 −3/2 (3.108) ds U (|r + s|) exp − 2 , Vij (r) = (2πσ ) 2σ where r = r ij = r i − r j , , σ2 = 12μkB T m1 m2 . μ = m1 + m2

(3.109) (3.110) (3.111)

This is the Feynman–Hibbs potential. It can be evaluated for any wellbehaved interaction function U (r) from the integral (we write z = cosine of the angle between r and s):

√ −3 ∞ 1 s2 FH 2 2 2 Vij (r) = (σ 2π) ds dz 2πs U ( r + s − 2rsz) exp − 2 . 2σ 0 −1 (3.112) Some insight is obtained by expanding U to second order in s/r. Using

1 s2 s s3 2 r2 + s2 − 2rsz = r 1 − z + (1 − z ) + O( ) , (3.113) r 2 r2 r3 √ U ( r2 + s2 − 2rsz) expands as

1 2 2 U (r) 2 U = U (r) − szU (r) + s (1 − z ) + z U (r) . (3.114) 2 r Evaluating the integral (3.112), we ﬁnd VijFH (r) 8

2 = U (r) + 24μkB T

2U (r) + U (r) . r

(3.115)

See Mak and Andersen (1990) and Cao and Berne (1990) for a discussion of reference potentials.

72

From quantum to classical mechanics: when and how

It is left to Exercises 3.2 and 3.3 to evaluate the practical importance of the potential correction. For applications to Lennard-Jones liquids see Ses´e (1992, 1993, 1994, 1995, 1996). Guillot and Guissani (1998) applied the Feynman–Hibbs approach to liquid water.

3.5.2 The Wigner correction to the free energy The Wigner 2 corrections to the classical canonical partition function Q and Helmholtz free energy A are treated in Section 17.6 with the ﬁnal result in terms of Q given in (17.102) on page 476. Summarizing it is found that Q = Qcl (1 + fcor ), A = A − kB T fcor ,

1 2 1 2 2 (∇i V ) . ∇i V − fcor = − 2T2 mi 2kB T 12 kB cl

(3.116) (3.117) (3.118)

i

The two terms containing potential derivatives can be expressed in each other when averaged over the canonical ensemble: (∇i V )2 = kB T ∇2i V ,

(3.119)

as we shall prove below. Realizing that the force F i on the i-th particle is equal to −∇i V , (3.118) can be rewritten as fcor = −

1 2 F 2 . 3T3 mi i 24 kB i

(3.120)

This is a convenient form for practical use. For molecules it is possible to split the sum of squared forces into translational and rotational degrees of freedom (see Powles and Rickayzen, 1979). These are potential energy corrections; one should also be aware of the often non-negligible quantum corrections to the classical rotational partition function, which are of a kinetic nature. For formulas the reader is referred to Singh and Sinha (1987), Millot et al. (1998) and Schenter (2002). The latter two references also give corrections to the second virial coeﬃcient of molecules, with application to water. Proof We prove (3.119). Consider one particular term, say the second derivative to x1 in ∇V : 2 ∂ V −βV e dx1 dr , ∂x21

3.5 Quantum corrections to classical behavior

73

where the prime means integration over all space coordinates except x1 . Now, by partial integration, we obtain x1 =∞ ∂V 2 −βV −βV ∂V e dr + e dr. ∂x1 x1 =−∞ ∂x1 The ﬁrst term is a boundary term, which vanishes for a ﬁnite system where the integrand goes to zero at the boundary if the latter is taken beyond all particles. It also vanishes for a periodic system because of the equality of the integrand at periodic boundaries. Since every term in ∇2 V can be equally transformed, (3.119) follows.

3.5.3 Equivalence between Feynman–Hibbs and Wigner corrections We now show that application of the Feynman-Hibbs potential (3.115) yields the same partition function and free energy as application of the Wigner correction (3.118). We start by rewriting (3.118), using (3.119): fcor = −

1 2 ∇2 V. 2T2 mi i 24 kB i

(3.121)

Assuming V can be written as a sum of pair potentials U (r): V =

U (rij ) =

i<j

1 U (rij ), 2

we can evaluate the Laplacian and arrive at 1 2U (rij ) 2 U (rij ) + . fcor = − 2T2 mi rij 24 kB i

(3.122)

i,j =i

(3.123)

j =i

Next we rewrite the total potential energy on the basis of Feynman–Hibbs pair potentials: V FH =

1 FH Vij (rij ) 2 i,j =i

2U (rij ) 1 1 2 1 = V + U (rij ) + + 2 24 kB T mi mj rij i,j =i 2U (rij ) 2 1 cl U (rij ) + . (3.124) = V + 24 kB T mi rij cl

i

j =i

74

From quantum to classical mechanics: when and how

Expanding exp(−βV FH ) to ﬁrst order in the correction term we ﬁnd: ⎤ ⎡ 2 2U (rij ) ⎦ 1 FH cl U (rij ) + , (3.125) e−βV = e−βV ⎣1 − 24 kB T mi rij i

j =i

which, after integration, gives exactly the fcor of (3.123).

3.5.4 Corrections for high-frequency oscillators Bond oscillations are often of such a high frequency that ω/kB T > 1 and order-2 corrections are not suﬃcient to describe the thermodynamics of the vibrations correctly. A good model for non-classical high-frequency vibrations is the harmonic oscillator, which is treated in Chapter 17. Figure 17.5 on page 478 shows the free energy for the harmonic oscillator for the classical case, the 2 -corrected classical case and the exact quantum case. When ω/kB T >≈ 5, the bond vibrational mode is essentially in its ground state and may be considered as ﬂexible constraint in simulations.9 For ω/kB T <≈ 2, the 2 -corrected values, which can be obtained by a proper Feynman–Hibbs potential, are quite accurate. For the diﬃcult range 2 < ω/kB T < 5 it is recommended to use the exact quantum corrections. At T = 300 K, this “diﬃcult” range corresponds to wave numbers between 400 and 1000 cm−1 in which many vibrations and librations occur in molecules. One may also choose not to include the 2 corrections at all and apply the exact quantum corrections for the full range ω/kB T >≈ 0.5, i.e., all frequencies above 100 cm−1 . In a condensed-phase simulation, one does not know all the normal modes of the system, from which the quantum corrections could be computed. The best way to proceed is to perform a classical simulation and compute the power spectrum of the velocities of the particles.10 The power spectrum will contain prominent peaks at frequencies corresponding to normal modes. We follow the description by Berens et al. (1983), who applied quantum corrections to water. First compute the total mass-weighted velocity-correlation function for an N -particle ﬂuid: C(τ ) =

3N

mi vi (t)vi (t + τ ),

(3.126)

i=1 9 10

See the treatment of ﬂexible constraints in molecular dynamics on page 158. See Section 12.8 for the description of power spectra and their relation to correlation functions.

3.5 Quantum corrections to classical behavior

and from this the spectral density of states S(ν): ∞ 4 C(τ ) cos 2πντ dτ. S(ν) = kB T 0

75

(3.127)

Note that the correlation function is the inverse transform of S(ν) (see Section 12.8): ∞ C(τ ) = kB T S(ν) cos 2πντ dν, (3.128) 0

with special case

∞

S(ν) dν = 0

C(0) = 3N. kB T

(3.129)

Now we have the classical density of states, we can compute the quantum corrections to thermodynamics quantities. The results are (Berens et al., 1983):

1 − e−ξ dνS(ν) ln −ξ/2 − ln ξ , e 0

∞ ξ ξ + ξ −1 , dνS(ν) 2 e −1 0 ∞

ξ −ξ , − ln 1 − e dνS(ν) ξ e −1 0

∞ ξ 2 eξ dνS(ν) −1 , (1 − eξ )2 0 qu

A

−A

cl

= kB T

U qu − U cl = kB T S qu − S cl = kB T CVqu − CVcl = kB T

∞

(3.130) (3.131) (3.132) (3.133)

where ξ=

hν . kB T

(3.134)

One should check if the computed spectral density of states integrate to 3N .

3.5.5 The fermion–boson exchange correction In Section 17.6 the classical approximation to quantum statistical mechanics has been derived. In Eq. (17.112) on page 479 a correction for the fermion or boson character of the particle is given in the form of a repulsive or attractive short-range potential. As shown in Fig. 17.6, the exchange correction potential for nuclei can be neglected in all but very exceptional cases.

76

From quantum to classical mechanics: when and how

Exercises 3.1 3.2 3.3

3.4

Check the expansions (3.113) and (3.114). Give an analytical expression for the Feynman-Hibbs potential in the approximation of (3.115) for a Lennard-Jones interaction. Evaluate both the full integral (3.112) and the approximation of the previous exercise for a He–He interaction at T = 40 K. Plot the results. Apply (3.120) to compute the partition function and the Helmholtz free energy of a system of N non-interacting harmonic oscillators and prove the correctness of the result by expanding the exact expression (from (17.84) on page 472).

4 Quantum chemistry: solving the time-independent Schr¨odinger equation

4.1 Introduction As has become clear in the previous chapter, electrons (almost) always behave as quantum particles; classical approximations are (almost) never valid. In general one is interested in the time-dependent behavior of systems containing electrons, which is the subject of following chapters. The time-dependent behavior of systems of particles spreads over very large time ranges: while optical transitions take place below the femtosecond range, macroscopic dynamics concerns macroscopic times as well. The light electrons move considerably faster than the heavier nuclei, and collective motions over many nuclei are slower still. For many aspects of long-time behavior the motion of electrons can be treated in an environment considered stationary. The electrons are usually in bound states, determined by the positions of the charged nuclei in space, which provide an external ﬁeld for the electrons. If the external ﬁeld is stationary, the electron wave functions are stationary oscillating functions. The approximation in which the motion of the particles (i.e., nuclei) that generate the external ﬁeld, is neglected, is called the Born–Oppenheimer approximation. Even if the external ﬁeld is not stationary (to be treated in Chapter 5), the non-stationary solutions for the electronic motion are often expressed in terms of the pre-computed stationary solutions of the Schr¨ odinger equation. This chapter concerns the computation of such stationary solutions. Thus, in this chapter, the Schr¨ odinger equation reduces to a time-independent problem with a stationary (i.e., still time-dependent, but periodic) solution. Almost all of chemistry is covered by this approximation. It is not surprising, therefore, that theoretical chemistry has been almost equivalent to quantum chemistry of stationary states, at least up to the 1990s, when 77

78

Quantum chemistry: solving the time-independent Schr¨ odinger equation

the scope of theory in chemistry slowly started to be broadened to include the study of more complex dynamic behavior. For completeness, in the last section of this chapter attention will be given to the stationary quantum behavior of nuclei, rather than electrons. This includes the rotational and vibrational steady state behavior of molecules, which is useful in spectroscopic (infrared and Raman) studies, in the prediction of spectroscopic behavior by simulations, or in the use of spectroscopic data to evaluate force ﬁelds designed for simulations. 4.2 Stationary solutions of the TDSE The general form of the time-dependent Schr¨ odinger equation (TDSE) is ∂Ψ ˆ = HΨ, (4.1) ∂t where the usual (already simpliﬁed!) form of the Hamiltonian is that of (2.72). If the Hamiltonian does not contain any explicit time dependence and is only a function of the particle coordinates and a stationary external potential, the TDSE has stationary solutions that represent bound states: i Ψn (r, t) = ψn (r) exp − En t , (4.2) i

where ψn (r) and En are solutions of the eigenvalue equation ˆ Hψ(r) = Eψ(r).

(4.3)

The latter is also called the time-independent Schr¨ odinger equation The spatial parts of the wave functions are stationary in time, and so is the probability distribution Ψ∗n Ψn for each state. In this chapter we shall look at ways to solve the time-independent Schr¨ odinger equation, (4.3), assuming stationary external ﬁelds. In chapter 5 we consider how a quantum system behaves if the external ﬁeld is not stationary, for example if the nuclei move as well, or if there are external sources for ﬂuctuating potentials. There are several ways in which ab initio solutions of the time-independent Schr¨ odinger equation can be obtained. In quantum physics the emphasis is often on the behavior of a number of quantum particles, which are either bosons, as in helium-4 liquids, or fermions as electrons in (semi)conductors or in helium-3 liquids. In chemistry the main concern is the structure and properties of single atoms and molecules; especially large molecules with many electrons pose severe computational problems and elude exact treatment.

4.3 The few-particle problem

79

Before considering methods to solve the many-electron problem, we shall look into the methods that are available to ﬁnd the stationary solution for one or a few interacting quantum particles. Then we consider the question whether it will be possible to separate the nuclear motion from the electronic motion in atoms and molecules: this separation is the essence of the Born–Oppenheimer approximation. When valid, the electronic motion can be considered in a stationary external ﬁeld, caused by the nuclei, while the nuclear motion can be described under the inﬂuence of an eﬀective potential caused by the electrons.

4.3 The few-particle problem Let us ﬁrst turn to simple low-dimensional cases. Mathematically, the SE is a boundary-value problem, with acceptable solutions only existing for discrete values of E. These are called the eigenvalues, and the corresponding solutions the eigenfunctions. The boundary conditions are generally zero values of the wavefunction at the boundaries,1 and square-integrability of the function, i.e., ψ ∗ ψ(x) dx must exist. As any multiple of a solution is also a solution, this property allows to normalize each solution, such that the integral of its square is equal to one. Since any Hamiltonian is Hermitian (see Chapter 14), its eigenvalues E are real. But most Hamiltonians are also real, except when velocity-dependent potentials as in magnetic interactions occur. Then, when ψ is a solution, also ψ ∗ is a solution for the same eigenvalue, and the sum of ψ and ψ ∗ is a solution as well. So the eigenfunctions can be chosen as real functions. Often, however, a complex function is chosen instead for convenience. For example, instead of working with the real functions sin mφ and cos mφ, one may more conveniently work with the complex functions exp(imφ) and exp(−imφ). Multiplying a wave function by a constant exp(ia) does not change any of the physical quantities derived from the wave function. Consider a single quantum particle with mass m in a given, stationary, external potential V (x). We shall not treat the analytical solutions for simple cases such as the hydrogen atom, as these can be found in any text book on quantum physics or theoretical chemistry. For the one-dimensional case there are several ways to solve the time-independent SE numerically. 1

In the case of periodic boundary conditions, the wave function and its derivatives must be continuous at the boundary.

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Quantum chemistry: solving the time-independent Schr¨ odinger equation

4.3.1 Shooting methods The popular shooting methods integrate the second-order diﬀerential equation 2m d2 ψ(x) = 2 [V (x) − E]ψ(x) (4.4) 2 dx from one end with an estimate of the eigenvalue E and iterate over changes of E; only when E is equal to an eigenvalue will the wave function fulﬁll the proper boundary condition at the other end. In fact, one replaces a boundary value problem with an iterative initial value problem.2 The Numerov method is recommended; it consists of solving (4.4) to fourth-order precision in the grid spacing, requiring the second derivative of the potential, which is given by (4.4) and which is discretized using three points. It is left to the reader as an exercise to derive the corresponding algorithm. Function numerov(m,E,V) ﬁnds the nearest eigenvalue by iteration, starting from a guessed value for E. It shoots from both sides to a common point, and after scaling, compares the ﬁrst derivatives at the end points (Pang, 1997). Then E is adjusted to equalize both relative derivatives with a linear inter-/extrapolation root search method. The shooting method is in principle exact (limited only by discretization errors) and has the advantage that it can also generate excited states, but it is not very suitable for higher dimensional cases. python program 4.1 numerov(m,E,V) Finds the nearest eigenvalue for the single-particle Schr¨ odinger equation. 01 def numerov(m,E,V): 02 # m=index for matching point 03 # E=trial energy*mass*delx**2/hbar**2 04 # V=array of pot. energy*mass*delx**2/hbar**2 05 # returns [nr of zerocrossings, difference in relat. first derivatives] 06 E1=E; E2=E1*1.05 07 F1=shoot(m,E1,V) 08 while (abs(E2-E1)> 1.e-8): 09 nF=shoot(m,E2,V); F2=nF[1] 10 Etemp=E2 11 E2=(F1*E2-F2*E1)/(F1-F2) 12 E1=Etemp 13 F1=F2 14 print ’%3d %13.10f’ %(nF[0], E2) 15 return [nF[0],E2] 16 2

The one-dimensional Schr¨ odinger equation is a special case of the class of Sturm–Liouville df (x) d problems: dx (p(x) dx ) + q(x)f (x) = s(x). See Pang (1997) or Vesely (2001) for a discussion of such methods in a more mathematical context. Both books describe Numerov’s method in detail.

4.3 The few-particle problem 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49

81

def shoot(m,E,V): # m=index of matching point, should be near right end # E=trial energy*mass*delx**2/hbar**2 # V=array of pot. energy*mass*delx**2/hbar**2 # returns [nr of zerocrossings, difference in first derivatives] nx=len(V) ypresent=0.; yafter=0.001 i=1 sign=1. nz=0 while i <= m: # shoot from left to right ybefore=ypresent; ypresent=yafter gplus=1.-(V[i+1]-E)/6. gmed=2.+(5./3.)*(V[i]-E) gmin=1.-(V[i-1]-E)/6. yafter=gmed*ypresent/gplus -gmin*ybefore/gplus if (yafter*sign < 0.): nz=nz+1 sign=-sign i=i+1 ym=ypresent forwardderiv=yafter-ybefore ypresent=0.; yafter=0.001 i=nx-2 while i >= m: #shoot from right to left ybefore=ypresent; ypresent=yafter gplus=1.-(V[i-1]-E)/6. gmed=2.+(5./3.)*(V[i]-E) gmin=1.-(V[i+1]-E)/6. yafter=gmed*ypresent/gplus -gmin*ybefore/gplus i=i-1 backwardderiv=(ybefore-yafter)*ym/ypresent return [nz,forwardder-backwardder]

Comments Line 02: m is the point where the forward and backward “shooting” should match. It is best taken near the right border, say at 80% of the length of the vectors. Line 06: the ﬁrst two guesses are E1 and E1 + 5%. Line 09: this produces the next guess, using the value of E produced in line 12 of the previous step, which is based on a linear relation between E and the output of shoot (diﬀerence between derivatives at matching point). The routine may not always converge to the expected (nearest) eigenvalue, as the output of shoot is very erratic when E deviates far from any eigenvalue. Line 15: The routine also returns the number of nodes in the wave function, which is an indication of the eigenvalue number. Note that numerov does not produce the wave function itself. In order to generate φ when E is already known, the function psi(m,E,V) can be called.

python program 4.2 psi(m,E,V) Constructs the wave function for a given exact eigenvalue from the single-particle Schr¨ odinger equation. 01 02 03 04 05 06

def psi(m,E,V): # m=index of matching point # E=energy*mass*delx**2/hbar**2 must be converged for same m and V # V=array of pot. energy*mass*delx**2/hbar**2 # returns wave function y; sum(y**2)=1 nx=len(V)

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Quantum chemistry: solving the time-independent Schr¨ odinger equation

07 08 09 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29

y=zeros(nx,dtype=float) y[1]=0.001 i=1 while i < m: # shoot from left to right gplus=1.-(V[i+1]-E)/6. gmed=2.+(5./3.)*(V[i]-E) gmin=1.-(V[i-1]-E)/6. y[i+1]=gmed*y[i]/gplus -gmin*y[i-1]/gplus i=i+1 ym=y[m] y[-2]=0.001 i=nx-2 sign=1 while i > m: #shoot from right to left gplus=1.-(V[i-1]-E)/6. gmed=2.+(5./3.)*(V[i]-E) gmin=1.-(V[i+1]-E)/6. y[i-1]=gmed*y[i]/gplus -gmin*y[i+1]/gplus i=i-1 scale=ym/y[m] for i in range(m,nx): y[i]=scale*y[i] y=y/sqrt(sum(y**2)) return y

Comments This algorithm is similar to the shoot algorithm, except that an array of y-values is kept and the number of zero crossings is not monitored. Lines 26–27 scale the backward shot on the forward one; line 28 normalizes the total sum of squares to 1.

Figure 4.1 illustrates the use of numerov and psi to compute the ﬁrst six eigenvalues and eigenfunctions for a double-well potential, exemplifying the proton potential energy in a symmetric hydrogen bond.3 The potential is composed of the sum of two opposing Morse potentials, each with D = 600 kJ/mol and a harmonic vibration frequency of 100 ps−1 (3336 cm−1 ), one with minimum at 0.1 nm and the other at 0.2 nm. The lowest two levels are adiabatic tunneling states, which diﬀer only slightly in energy (0.630 kJ/mol) with wave functions that are essentially the sum and diﬀerence of the diabatic ground state wave functions of the individual wells. In each of the adiabatic states the proton actually oscillates between the two wells with a frequency of (E1 − E0 )/h = 1.579 THz; it tunnels through the barrier. The excited states all lie above the energy barrier; the proton then has enough energy to move over the barrier.

4.3.2 Expansion on a basis set Another class of solutions is found by expanding the unknown solution in a ﬁnite number of properly chosen basis functions φn : 3

See, e.g., Mavri and Grdadolnik (2001), who ﬁt two Morse potentials plus modifying terms to high-level quantum calculations in the case of acetyl acetone.

4.3 The few-particle problem

83

V,E (kJ/mol) 70 60 50 40 30 20 10 0 0.1

0.125

0.15

0.175

0.2 xproton (nm)

Figure 4.1 Lowest six proton quantum states in a double-well potential (thick curve), typical for a proton in a symmetric hydrogen bond. Levels and wave functions were generated with the shooting method (see text). Wave functions are indicated by alternating solid and dotted thin lines. The energies of the states are (in kJ/mol): 0:10.504, 1:11.135, 2:25.102, 3:32.659, 4:45.008, 5:58.804.

ψ(x) =

cn φn (x).

(4.5)

n

The time-independent Schr¨ odinger equation now becomes an eigenvalue equation (see Chapter ??): Hc = λSc, where S is the overlap matrix Snm =

φ∗n φm dx.

(4.6)

(4.7)

For an orthogonal basis set the overlap matrix is the unit matrix. The eigenvalue equation can be solved by standard methods (see, e.g., Press et al., 1992). It is most convenient to diagonalize the basis set ﬁrst,4 which 4

Diagonalization is not a unique procedure, as there are more unknown mixing coeﬃcients than equations. For example, mixing two functions requires four coeﬃcients, while there are three conditions: two normalizations and one zero overlap. One unique method is to start with

84

Quantum chemistry: solving the time-independent Schr¨ odinger equation

is a one-time operation after which the eigenvalue problems become much easier. Such methods do yield excited states as well, and are extendable to higher dimensions, but they are never exact: their accuracy depends on the suitability of the chosen basis set. Still, the solution of many-electron problems as in quantum chemistry depends on this approach. An example of the use of a basis set can be found in the study of proton transfer over hydrogen bonds (Berendsen and Mavri, 1997). For solving the time-dependent SE (see Chapter 5) a description of the proton wave function for a ﬂuctuating double-well potential in terms of a simple basis set was needed. The simplest basis set that can describe the tunneling process consists of two Gaussian functions, each resembling the diabatic ground state solution of one well (Mavri et al., 1993). Many analytical theories rely on such two-state models. It turns out, however, that a reasonable accuracy can only be obtained with more than two Gaussians; ﬁve Gaussians, if properly chosen, can reproduce the ﬁrst few eigenstates reasonably well (Mavri and Berendsen, 1995).

4.3.3 Variational Monte Carlo methods The variational method consists of deﬁning some ﬂexible function with a number of adjustable parameters that is expected to encompass a good approximation to the true wave function. The variational principle says that the expectation of the Hamiltonian over any function ψ(r) (which must be quadratically integrable) is always larger than or equal to the ground state energy with equality only when the function is identical to the ground state eigenfunction: ˆ ˆ ψ(r)Hψ(r) dr Hψ = ≥ E0 (4.8) E = ψ ψ 2 (r) 2 ψ

ˆ yield the Therefore the parameter values that minimize the expectation of H best approximation to the ground state wave function. For low-dimensional problems and for linear dependence on the parameters, the integral can in general be evaluated, and the minimization achieved. However, for multidimensional cases and when the integral cannot be split up into a linear combination of computable components, the multidimensional integral is better one normalized eigenfunction, say φ1 , then make φ2 orthogonal to φ1 by mixing the right amount of φ1 into it, then normalizing φ2 , and proceeding with φ3 in a similar way, making it orthogonal to both φ1 and φ2 , etc. This procedure is called Schmidt orthonormalization, see, e.g., Kyrala (1967). A more symmetrical result is obtained by diagonalizing the overlap matrix of normalized functions by an orthogonal transformation.

4.3 The few-particle problem

85

solved by Monte Carlo techniques. When an ensemble of conﬁgurations is generated that is representative for the probability distribution ψ 2 , the inˆ tegral is approximated by the ensemble average of Hψ/ψ, which is a local property that is usually easy to determine. The parameters are then varied to minimize E. The trial wave function may contain electron–electron correlation terms, for example in the Jastrow form of pair correlations, and preferably also three-body correlations, while it must fulﬁll the parity requirements for the particles studied. The generation of conﬁgurations can be done as follows. Assume a starting conﬁguration r 1 with ψ 2 (r 1 ) = P1 is available. The “local energy” for the starting conﬁguration is ˆ Hψ(r 1) ε1 = . (4.9) ψ(r 1 ) (i) Displace either one coordinate at the time, or all coordinates simultaneously, with a random displacement homogeneously distributed over a given symmetric range (−a, a). The new conﬁguration is r 2 . (ii) Compute P2 = ψ 2 (r 2 ). (iii) If P2 ≥ P1 , accept the new conﬁguration; if P2 < P1 , accept the new conﬁguration with probability P2 /P1 . This can be done by choosing a random number η between 0 and 1 and accept when η ≤ P2 /P1 . (iv) If the move is accepted, compute the “local energy” ε2 =

¯ Hψ(r 2) ; ψ(r 2 )

(4.10)

if the move is rejected, count the conﬁguration r 1 and its energy ε1 again; (v) Repeat steps (i)–(iv) N times. (vi) The expectation of the Hamiltonian is the average energy N εi ˆ H = i=1 . (4.11) N The range for the random steps should be chosen such that the acceptance ratio lies in the range 40 to 70%. Note that variational methods are not exact, as they depend on the quality of the trial function.

4.3.4 Relaxation methods We now turn to solutions that make use of relaxation towards the stationary solution in time. We introduce an artiﬁcial time dependence into the wave

86

Quantum chemistry: solving the time-independent Schr¨ odinger equation

function ψ(x, τ ) and consider the partial diﬀerential equation ∂ψ 2 V −E = ∇ ψ− ψ. (4.12) ∂τ 2m It is clear that, if ψ equals an eigenfunction and E equals the corresponding eigenvalue of the Hamiltonian, the right-hand side of the equation vanishes and ψ will not change in time. If E diﬀers from the eigenvalue by ΔE, the total magnitude of ψ (e.g., the integral over ψ 2 ) will either increase or decrease with time: ΔE dI = I, I = ψ 2 dx. (4.13) dτ So, the magnitude of the wave function is not in general conserved. If ψ is not an eigenfunction, it can be considered as a superposition of eigenfunctions φn : ψ(x, τ ) = cn (τ )φn (x). (4.14) n

Each component will now behave in time according to E − En dcn = cn , dτ or

cn (τ ) = cn (0) exp

E − En τ .

(4.15)

(4.16)

This shows that functional components with high eigenvalues will decay faster than those with lower eigenvalues; after suﬃciently long time only the ground state, having the lowest eigenvalue, will survive. Whether the ground state will decay or grow depends on the value chosen for E and it will be possible to determine the energy of the ground state by monitoring the scaling necessary to keep the magnitude of ψ constant. Thus the relaxation methods will determine the ground state wave function and energy. Excited states can only be found by explicitly preventing the ground state to mix into the solution; e.g., if any ground state component is consistently removed during the evolution, the function will decay to the ﬁrst excited state. Comparing (4.12), setting E = 0, with the time-dependent Schr¨ odinger equation (4.1), we see that these equations are equivalent if t is replaced by iτ . So, formally, we can say that the relaxation equation is the TDSE in imaginary time. This sounds very sophisticated, but there is no deep physics behind this equivalence and its main function will be to impress one’s friends! As an example we’ll generate the ground state for the Morse oscillator

4.3 The few-particle problem

87

(see page 6) of the HF molecule. There exists an analytical solution for the Morse oscillator:5 (ω0 )2 1 1 (n + )2 , En = ω0 (n + ) − 2 4D 2 yielding

(4.17)

ω0 1 − , E0 = ω0 2 16D 3 9ω0 − . E1 = ω0 2 16D

(4.18) (4.19)

For HF (see Table 1.1) the ground state energy is 24.7617 kJ/mol for the harmonic approximation and 24.4924 kJ/mol for the Morse potential. The ﬁrst excited state is 74.2841 kJ/mol (h.o.) and 71.8633 kJ/mol (Morse). In order to solve (4.12) ﬁrst discretize the distance x in a selected range, with interval Δx. The second derivative is discretized as ψi−1 − 2ψi + ψi+1 ∂2ψ = . ∂x2 (Δx)2

(4.20)

If we choose Δτ =

m(Δx)2 ,

def

m =

mH + mF , mH mF

(4.21)

then we ﬁnd that the second derivative leads to a computationally convenient change in ψi : ψi (τ + Δτ ) = 12 ψi−1 (τ ) + 12 ψi+1 (τ ).

(4.22)

Using a table of values for the Morse potential at the discrete distances, multiplied by Δτ / and denoted below by W , the following Python function will perform one step in τ . python program 4.3 SRstep(n,x,y,W) Integrates one step of single-particle Schr¨ odinger equation in imaginary time. 01 def SRstep(x,y,W): 02 # x=distance array; 03 # y=positive wave function; sum(y)=1 required; y[0]=y[1]=y[-2]=y[-1]=0. 04 # W=potential*delta tau/hbar 05 # returns wave function and energy*delta tau/hbar 06 z=concatenate(([0.],0.5*(y[2:]+y[:-2]),[0.])) 07 z[1]=z[-2]=0. 5

See the original article of Morse (1929) or more recent texts as Levin (2002); for details and derivation see Mathews and Walker (1970) or Fl¨ ugge (1974).

88

Quantum chemistry: solving the time-independent Schr¨ odinger equation

V (kJ/mol) x103 1

0.8 wave function first excited state

dissociation energy

0.6 wave function ground state

0.4

potential energy

0.2 first excited state ground state level

0.05

0.1

0.15

0.2

0.25

0.3

rHF (nm) Figure 4.2 The Morse potential for the vibration of hydrogen ﬂuoride and the solution for the ground and ﬁrst excited state vibrational levels (in the absence of molecular rotation), obtained by relaxation in imaginary time.

08 09 10 11 12

z=z*exp(-W) s=sum(z) E=-log(s) y=z/s return [y,E]

Comments Line 03: the ﬁrst two and last two points are zero and are kept zero. Line 06: diﬀusion step: each point becomes the average of its neighbors; zeros added at ends. Line 07: second and before-last points set to zero. Line 08: evolution due to potential. Line 10: Energy (∗Δτ /) needed to keep y normalized. This converges to the ground state energy. Line 12: Use y = SRstep[0] as input for next step. Monitor E = step[1] every 100 steps for convergence. Last y is ground state wave function.

Using the values for HF given in Table 1.1, using 1000 points (0, 3b) (b is the bond length), and starting with a Gaussian centered around b with σ = 0.01 nm, the energy value settles after several thousand iterations to 24.495 kJ/mol. The resulting wave function is plotted in Fig. 4.2. It very closely resembles the Gaussian expected in the harmonic approximation. The algorithm is only of second-order accuracy and the discretization error is proportional to (Δx)2 , amounting to some 3 J/mol for 1000 points.

4.3 The few-particle problem

89

The ﬁrst excited state can be computed when the wave function is kept orthogonal to the ground state. Denoting the ground state by ψ0 , we add a proportion of ψ0 to ψ such that (4.23) ψψ0 dx = 0, which can be accomplished by adding a line between line 8 and 9 in the program SRstep: 08a z=z-sum(z*y0)*y0 where y0 is the converged and normalized ground state wave function. Figure 4.2 includes the ﬁrst excited state, with energy 71.866 kJ/mol. This is accurate within 3 J/mol. The vibration wave numbers then are 4139.8 (harmonic oscillator) and 3959.9 (Morse function). Higher excited states can be computed as well, as long as the wave function is kept orthogonal to all lower state wave functions.

4.3.5 Diﬀusional quantum Monte Carlo methods If we look carefully at (4.12), we see that the ﬁrst term on the r.h.s. is a diﬀusion term, as in Fick’s equation for the time dependence of the concentration c of diﬀusing particles: ∂c = D∇2 c. ∂t

(4.24)

This ﬁts with what this term does after discretization (4.22): the function splits in two halves located at the neighboring points. This is what would happen with a probability distribution of particles after each particle has made a random step over one grid element, either to the left or to the right. This equivalence suggests a diﬀerent way to solve the SE, which has been pioneered by Anderson (1975, 1976).6 Suppose we wish to obtain the ground state of a system of n particles. Consider an ensemble of a large number of replicas of the system, each with its own conﬁguration of the n particles. The members of the ensemble are called psi-particles (psips) or walkers. Then evolve each member of the ensemble as follows: (i) Displace each of the coordinates of the particles in a “time” Δτ with a random displacement with variance (Δx)2 = 2DΔτ = 6

Δτ. m

(4.25)

See reviews by Anderson (1995) and by Foulkes et al. (2001), the latter with applications to solids.

90

Quantum chemistry: solving the time-independent Schr¨ odinger equation

This can be done by sampling from a Gaussian distribution with that variance, or by displacing the coordinate by ± Δτ /m. (ii) Duplicate or annihilate the walker, according to the probability P = exp[−(V − E)Δτ /]

(4.26)

in order to satisfy the second term (source term) on the r.h.s. of (4.12). This, again, can be done in several ways. One way is to let the walker survive with a probability P if P < 1, and let it survive but create a new walker with probability P − 1 if P ≥ 1. This is accomplished (Foulkes et al., 1995) by deﬁning the new number of walkers Mnew as Mnew = integer(P + η),

(4.27)

where η is a uniform random number between 0 and 1. A higherorder accuracy is obtained when the potential energy for the creation/annihilation (in (4.26)) is taken as the average before and after the diﬀusion step: V = 12 [V (τ ) + V (τ + Δτ )].

(4.28)

Another way is to enhance or reduce a weight per walker, and then applying a scheme like: duplication when the weight reaches 2 or annihilation with 50% probability when the weight falls below 0.5; the surviving and new walkers will then start with weight 1. Applying weights only without creation/annihilation scheme does not work, as this produces uneven distributions with dominating walkers. (iii) The energy E determines the net gain or loss of the number of walkers and should be adjusted to keep that number stationary. The advantage of such a stochastic scheme above relaxation on a grid is that it can more easily be expanded to several dimensions. For example, a four-particle system in 3D space (a hydrogen molecule) involves 12 degrees of freedom, reducible to six internal degrees of freedom by splitting-oﬀ overall translation and rotation. A grid with only 100 divisions per dimension would already involve the impossible number of 1012 grid points, while an ensemble of 105 to 106 walkers can do the job. The method remains exact, and includes all correlations between particles. The result has a statistical error that reduces with the inverse square root of the number of steps. Diﬀusional Monte Carlo methods cannot straightforwardly handle wave functions with nodes, which have positive as well as negative regions. Nodes occur not only for excited states, but also for ground states of systems

4.3 The few-particle problem

91

containing more than two identical fermions, or even with two fermions with the same spin. Only two fermions with opposite spin can occupy the same all-positive ground state wave function. The diﬀusing particles represent either positive or negative wave functions, and when a positive walker crosses a nodal surface, it would in principle cancel out with the negative wave function on the other side. One scheme that avoids these diﬃculties is to keep the nodal surface ﬁxed and annihilate the walkers that diﬀuse against that surface; however, by ﬁxing the nodal surface the method is no longer exact. Schemes that use exact cancelation between positive and negative walkers, instead of ﬁxed nodal surfaces, have been successfully applied by Anderson (1995). The implementation of diﬀusional Monte Carlo methods is much more eﬃcient when an approximate analytical wave function is used to guide the random walkers into important parts of conﬁguration space. This importance sampling was introduced by Grimm and Storer (1971) and pursued by Ceperley and Alder (1980). Instead of sampling ψ, the function f (r, τ ) = ψ(r, τ )ψT (r) is sampled by the walkers, where ψT is a suitable trial function. The trial function should be close to the real solution and preferably have the same node structure as the exact solution; in that case the function f is everywhere positive. Noting that the exact energy E is the ˆ for the solution ψ at inﬁnite time eigenvalue of H ˆ = Eψ Hψ

(4.29)

ˆ is Hermitian: and that H ˆ ˆ ψ HψT dr = ψT Hψ dr = E ψT ψ dr,

(4.30)

we can write the energy as the expectation of the local energy for the trial wave function (which can be evaluated for every member of the generated ensemble) over an ensemble with weight f = ψψT : Hψ ˆ ˆ T dr ˆ T f ψTT dr ψ Hψ Hψ = = E= . (4.31) ψT ψT ψ dr f dr f

This has the advantage that the energy follows from an ensemble average instead of from a rate of change of the number of walkers. The time-dependent equation for f follows, after some manipulation, from the deﬁnition of f and the Schr¨ odinger equation in imaginary time (4.12: ˆ T ∂f 1 Hψ 2 − E f. (4.32) = [∇ f − 2∇ · (f ∇ ln ψT )] − ∂τ 2m ψT

92

Quantum chemistry: solving the time-independent Schr¨ odinger equation

Note that in the multiparticle case the mass-containing term must be summed over all particles. The ﬁrst term on the r.h.s. of this equation is a diﬀusion term; the second term is a drift term: it is (minus) the divergence of a ﬂux density f u with a drift velocity u=

∇ ln ψT . m

(4.33)

The term − ln ψT acts as a guiding potential, steering the walker in the direction of large values of ψT . The third term replaces the strongly varying potential V (r) by the much weaker varying local energy for the trial function. QMC algorithms of this type suﬀer from a time-step error and extrapolation to zero time step is needed for a full evaluation. The accuracy can be considerably improved, allowing larger time steps, by inserting an acceptance/rejection step after the diﬀusion-drift step has been made (Reynolds et al., 1982; Umrigar et al., 1993). The procedure without the reactive step should lead to sampling of ψT2 , which can be made exact (for any time step) by accepting/rejecting or reweighting moves such as to maintain detailed balance under the known distribution ψT2 . 4.3.6 A practical example Let us, for the sake of clarity, work out the programming steps for a realistic case: the helium atom. We shall use atomic units (see page xvii) for which the electron mass, elementary charge, and are all unity. The helium atom consists of three particles: a nucleus with mass M (equal to 6544 for He-4), charge +2 and coordinates R, and two electrons with mass 1, charge −1 and coordinates r 1 , r 2 . The Hamiltonian, with nine degrees of freedom, reads 2 2 ˆ = − 1 ∇2 − 1 ∇2 − 1 ∇2 − H − + 2M R 2 1 2 2 |r 1 −R| |r 2 −R|

1 r12 ,

(4.34)

where r12 = |r 1 − r 2 |.

(4.35)

In the Born–Oppenheimer approximation we can eliminate R as a variable by assuming M = ∞ and R ≡ 0, but there is no pressing need to do so. Neither is there a need to separate center-of-mass and rotational coordinates and reduce the number of internal degrees of freedom to the possible minimum of three, and rewrite the Hamiltonian. We can simply use all nine coordinates; the eﬀect of reduced masses using internal coordinates is implied through the ﬁrst term concerning the diﬀusion of the heavy nucleus. The center-of-mass motion (corresponding to a free particle) will now also

4.3 The few-particle problem

93

be included, and will consist of a simple random walk, leading to an indefinitely expanding Gaussian distribution. All relevant data will concern the relative distribution of the particles. The attractive Coulomb terms in the Hamiltonian cause troublesome behavior when electrons move close to the nucleus. The time step should be taken very small in such conﬁgurations and it becomes virtually impossible to gather suﬃcient statistics. However, the use of importance sampling with a trial function that is the product of the single-electron ground-state solutions for the two electrons eliminates these attractive Coulombic terms altogether. Take as trial function ψT = exp[−α|r 1 − R| − α|r 2 − R|].

(4.36)

Choosing α=

2M , M +1

(4.37)

we ﬁnd that ˆ T 1 4M Hψ + =− . ψT M + 1 r12

(4.38)

We note that this trial function is a very rough approximation to the real wave function. For realistic applications it is necessary to use much better trial functions, e.g., obtained from variational Monte Carlo or density functional theory (see Section 4.7). The time evolution of f according to (4.32) is solved by a collection of walkers, each consisting of a nucleus and two electrons, that: (i) diﬀuse with a diﬀusion constant 1/2M , resp. 12 ; (ii) drift with a drift velocity (see (4.33)) r1 − R r2 − R α + u0 = M |r 1 − R| |r 2 − R|

(4.39)

for the nucleus, and ui = −α

ri − R |r i − R|

for the two electrons i = 1, 2; (iii) create/annihilate according to ∂f 4M 1 = E+ − f. ∂τ M + 1 r12

(4.40)

(4.41)

94

Quantum chemistry: solving the time-independent Schr¨ odinger equation

If the creation/annihilation step is implemented at each time by a stochastic process according to (4.27), additional noise is introduced into the process. It is better to assign a weight to each walker and readjust the weights every step, while some form of population control is carried out at regular intervals.7 The latter may involve duplication of heavy and annihilation of light walkers (according to Grassberger, 2002: if the weight exceeds a given upper threshold, then duplicate, giving each copy half the original weight; if the weight is less than a given lower threshold, draw a random number η between 0 and 1, annihilate if η < 12 , but keep with the double weight if η ≥ 12 ), or a complete random reassignment of walkers chosen from the weighted distribution of the existing walkers. In the following Python program a number of functions are deﬁned to realize the initial creation of walkers, the drift-diﬀusion step with readjustments of weights, and the population control. It is left to the reader to employ these functions in a simple program that computes the ground state energy of the helium-4 atom. There are two diﬀerent ways to compute the energy in excess of −4M/(M + 1): ﬁrst by monitoring the factor by which the weights must be readjusted to keep their sum constant (E), and second the average of 1/r12 over the ensemble of walkers (V ). When the time step is small enough, both energies tend to be equal; their diﬀerence is a good indicator for the suitability of the time step. One may choose 1000 walkers, a time step of 0.002 and make 1000 steps with weight adjustment before the walkers are renewed. The excess energy above the value of −3.999 455 hartree for −4M/(M + 1) should be recovered by E or V ; this value should equal +1.095 731, given the exact energy of the helium atom of −2.903 724 (Anderson et al., 1993). With this simple approach one may reach this value within 0.01 hartree, much better than the Hartree–Fock limit (see page 101) which is 0.04 hartree too high due to lack of correlation (Clementi and Roetti, 1974). python program 4.4 walkers Three functions to be used for simple QMD of the helium atom 01 def initiate(N): 02 # create array of N walkers (helium atom) 03 # returns [walkers,weights] 04 walkers=zeros((N,3,3), dtype=float) 05 sigma=0.5 06 walkers[:,1,:]=sigma*randn(N,3) 07 walkers[:,2,:]=sigma*randn(N,3) 7

See Hetherington (1984), Sorella (1998) and Assaraf et al. (2000) for a discussion of noise and bias related to stochastic reconﬁguration.

4.3 The few-particle problem 08 09 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48

95

r1=walkers[:,1]-walkers[:,0]; r2=walkers[:,2]-walkers[:,0] r1sq=sum(r1**2,1); r2sq=sum(r2**2,1) weights=exp(-2.*(sqrt(r1sq)+sqrt(r2sq))+0.5*(r1sq+r2sq)/sigma**2) ratio=N/sum(weights) weights=ratio*weights return [walkers,weights] def step(walkers,weights,delt,M): # move walkers one time step delt; M=nuclear/electron mass N=len(walkers) r1=walkers[:,1]-walkers[:,0]; r2=walkers[:,2]-walkers[:,0] r1norm=sqrt(sum(r1**2,1)); r2norm=sqrt(sum(r2**2,1)) for i in range(3): r1[:,i]=r1[:,i]/r1norm; r2[:,i]=r2[:,i]/r2norm alphadelt=2.*M/(M+1.)*delt d1=-alphadelt*r1; d2=-alphadelt*r2 d0=-(d1+d2)/M sd0=sqrt(delt/M); sd1=sqrt(delt) walkers[:,0,:]=walkers[:,0,:]+d0+sd0*randn(N,3) walkers[:,1,:]=walkers[:,1,:]+d1+sd1*randn(N,3) walkers[:,2,:]=walkers[:,2,:]+d2+sd2*randn(N,3) # adjust weights one time step V=1./sqrt(sum((walkers[:,1]-walkers[:,2])**2,1)) weights=weights*exp(-V*delt) ratio=N/sum(weights) E=log(ratio)/delt weights=ratio*weights return [walkers,weights,E] def renew(walkers,weights,Nnew): # select Nnew new walkers with unit weight wtacc=cumsum(weights) s=wtacc[-1] index=[] for i in range(Nnew): u=s*rand() arg=argmax(where(greater(wtacc,u),0.,wtacc)) + 1 index=index+[arg] wa=take(walkers,index) wt=ones((Nnew)) return [wa,wt]

Comments The coordinates of the walkers form an array walkers[n, i, j], where n numbers the walkers, i numbers the particles (0 = nucleus, 1,2 = electrons) in each walker and j = 0, 1, 2 indicates the x, y, z coordinates of a particle. Function initiate creates a number of N walkers: lines 06 and 07 assign a normal distribution to the electron coordinates (randn generates an array of normally distributed numbers); line 10 adjusts the weights to make the distribution exponential, and lines 11 and 12 normalize the total weight to N . The function step moves the particles in lines 26–28 by a simultaneous diﬀusion and drift displacement through sampling a normal distribution with prescribed mean and standard deviation. Then the weights are adjusted according to the computed excess energy 1/r12 of each walker in lines 30–31, and restored in lines 32–34 to the original total weight, which yields the “energy” E. The function renew reconstructs a new set of walkers each with unit weight, representing the same distribution as the old set. It ﬁrst constructs the cumulative sum of the original weights (in line 39); then, for each new walker, a random number, uniformly distributed over the total weight, is selected, and the index of the original walker in whose range the random number falls, is determined. All indices are appended in one list index (line 45) which is used in line 46 to copy the walkers that correspond to the elements of that list.

96

Quantum chemistry: solving the time-independent Schr¨ odinger equation

4.3.7 Green’s function Monte Carlo methods The diﬀusional and drift step of a walker can be considered as random samples from the Green’s function of the corresponding diﬀerential equation. The Green’s function G(r, τ ; r 0 , 0) of a linear, homogeneous diﬀerential equation is the solution ψ(r, τ ) when the boundary or initial condition is given by a delta-function ψ(r, 0) = δ(r − r 0 ); the general solution is the integral over the product of the Green’s function and the full boundary function: ψ(r, τ ) =

G(r, τ ; r 0 , 0)ψ(r 0 , 0) dr 0 .

(4.42)

For the diﬀusion equation the Green’s function is a Gaussian. There is an alternative, iterative way to solve the time-independent Schr¨ odinger equation by Monte Carlo moves, by iterating ψn according to 2m 2m 2 −∇ − 2 E ψn+1 = 2 V ψn . (4.43) The function ψn is sampled, again, by walkers, who make a step sampled from the Greens function of the diﬀerential operator on the left-hand-side, which in this case involves a modiﬁed Bessel function of the second kind. The iterations converge to the “exact” solution. We shall not further pursue these Green’s function Monte Carlo methods, which were originally described by Kalos (1962), and refer the reader to the literature.8 4.3.8 Some applications Quantum Monte Carlo methods have been used to solve several few-particle and some many-particle problems.9 Particular attention has been paid to the full potential energy surface of H3 in the Born–Oppenheimer approximation: a nine-dimensional problem (Wu et al., 1999). In such calculations involving a huge number of nuclear conﬁgurations, one can take advantage of the fact that the ﬁnal distribution of walkers for one nuclear conﬁguration is an eﬃcient starting distribution for a diﬀerent but nearby conﬁguration. All electron correlation is accounted for, and an accuracy of better than 50 J/mol is obtained. This accuracy is of the same order as the relativistic correction to the energy, as calculated for H2 (Wolniewicz, 1993). However, the adiabatic error due to the Born–Oppenheimer approximation is of the order of 1 kJ/mol and thus not all negligible. 8

9

See also Anderson (1995); three earlier papers, Ceperley and Kalos (1979), Schmidt and Kalos (1984) and Schmidt and Ceperley (1992), give a comprehensive review of quantum Monte Carlo methods. Schmidt (1987) gives a tutorial of the Green’s function Monte Carlo method. See Anderson (1976) and the references quoted in Anderson (1995).

4.4 The Born–Oppenheimer approximation

97

There is room for future application of QMC methods for large systems (Foulkes et al., 2001; Grossman and Mitas, 2005). Systems with up to a thousand electrons can already be treated. Trial wave functions can be obtained from density functional calculations (see below) and the QMC computation can be carried out on the ﬂy to provide a force ﬁeld for nuclear dynamics (Grossman and Mitas, 2005). Because QMC is in principle exact, it provides an ab initio approach to integrated dynamics involving nuclei and electrons, similar to but more exact than the “ab initio” molecular dynamics of Car and Parrinello (see Section 6.3.1). But even QMC is not exact as it is limited by the Born–Oppenheimer approximation, and because the nodal structure of the trial wave function is imposed on the wave function. The latter inconsistency may result in an incomplete recovery of the electron correlation energy, estimated by Foulkes et al. (2001) as some 5% of the total correlation energy. For elements beyond the second row of the periodic table, the replacement of core electrons by pseudopotentials becomes desirable for reasons of eﬃciency, which introduces further inaccuracies. Finally, we note that a QMC program package “Zori” has been developed by a Berkeley group of scientists, which is available in the public domain (Aspuru-Guzik et al., 2005).10 4.4 The Born–Oppenheimer approximation The Born–Oppenheimer (B–O) approximation is an expansion of the behavior of a system of nuclei and electrons in powers of a quantity equal to the electron mass m divided by the (average) nuclear mass M . Born and Oppenheimer (1927) have shown that the expansion should be made in (m/M )1/4 ; they also show that the ﬁrst and third order in the expansion vanish. The zero-order approximation assumes that the nuclear mass is inﬁnite, and therefore that the nuclei are stationary and their role is reduced to that of source of electrical potential for the electrons. This zero-order or clamped nuclei approximation is usually meant when one speaks of the B–O approximation per se. When nuclear motion is considered, the electrons adjust inﬁnitely fast to the nuclear position or wave function in the zero-order B–O approximation; this is the adiabatic limit. In this approximation the nuclear motion causes no changes in the electronic state, and the nuclear motion – both classical and quantum-mechanical – is governed by an eﬀective internuclear potential resulting from the electrons in their “stationary” state. The eﬀect of the adiabatic approximation on the energy levels of the 10

Internet site: http://www.zori-code.com.

98

Quantum chemistry: solving the time-independent Schr¨ odinger equation

hydrogen atom (where eﬀects are expected to be most severe) is easily evaluated. Disregarding relativistic corrections, the energy for a single electron atom with nuclear charge Z and mass M for quantum number n = 1, 2, . . . equals E=−

1 2μ hartree, Z 2n2 m

(4.44)

where μ is the reduced mass mM/(m + M ). All energies (and hence spectroscopic frequencies) scale with μ/m = 0.999 455 679. For the ground state of the hydrogen atom this means: E(adiabatic) = −0.500 000 000 hartree, E(exact) = −0.499 727 840 hartree, adiabatic error = −0.000 272 160 hartree , = −0.714 557 kJ/mol. Although this seems a sizeable eﬀect, the eﬀect on properties of molecules is small (Handy and Lee, 1996). For example, since the adiabatic correction to H2 amounts to 1.36 kJ/mol, the dissociation energy D0 of the hydrogen molecule increases by only 0.072 kJ/mol (on a total of 432 kJ/mol). The bond length of H2 increases by 0.0004 a.u. or 0.0002 ˚ Aand the vibrational −1 −1 frequency (4644 cm ) decreases by about 3 cm . For heavier atoms the eﬀects are smaller and in all cases negligible. Handy and Lee (1996) conclude that for the motion of nuclei the atomic masses rather than the nuclear masses should be taken into account. This amounts to treating the electrons as “following” the nuclei, which is in the spirit of the BO-approximation. The real eﬀect is related to the quantum-dynamical behavior of moving nuclei, especially when there are closely spaced electronic states involved. Such eﬀects are treated in the next chapter.

4.5 The many-electron problem of quantum chemistry Traditionally, the main concern of the branch of theoretical chemistry that is called quantum chemistry is to ﬁnd solutions to the stationary Schr¨ odinger equation for a system of (interacting) electrons in a stationary external ﬁeld. This describes isolated molecules in the Born–Oppenheimer approximation. There are essentially only two radically diﬀerent methods to solve Schr¨ odinger’s equation for a system of many (interacting) electrons in an external ﬁeld: Hartree–Fock methods with reﬁnements and Density Functional Theory (DFT). Each requires a book to explain the details (see Szabo and

4.6 Hartree–Fock methods

99

Ostlund, 1982; Parr and Yang, 1989), and we shall only review the principles of these methods. Statement of the problem We have N electrons in the ﬁeld of M point charges (nuclei). The point charges are stationary and the electrons interact only with electrostatic Coulomb terms. The electrons are collectively described by a wave function, which is a function of 3N spatial coordinates r i and N spin coordinates ωi , which we combine into 4N coordinates xi = r i , ωi . Moreover, the wave function is antisymmetric for exchange of any pair of electrons (parity rule for fermions) and the wave functions are solutions of the time-independent Schr¨ odinger equation: ˆ = EΨ HΨ

(4.45)

Ψ(x1 , . . . , xi , . . . , xj , . . . , ) = −Ψ(x1 , . . . , xj , . . . , xi , . . . , ), ˆ =− H

N N M 2 2 zk e2 ∇i − + 2m 4πε0 rik i=1

i=1 k=1

N i,j=1;i<j

e2 . 4πε0 rij

(4.46) (4.47)

By expressing quantities in atomic units (see page xvii), the Hamiltonian becomes zk 1 ˆ = −1 ∇2i − + . (4.48) H 2 rik rij i

i

k

i<j

ˆ is real, which implies that Ψ can be chosen to be real (if Ψ is Note that H a solution of (4.45), then Ψ∗ is a solution as well for the same energy, and so is (Ψ).) 4.6 Hartree–Fock methods The Hartree–Fock description of the wave function is in terms of products of one-electron wave functions ψ(r) that are solutions of one-electron equations (what these equations are will be described later). The one-electron wave functions are built up as a linear combination of spatial basis functions: ψi (r) =

K

cμi φμ (r).

(4.49)

μ=1

If the set of spatial basis functions would be complete (requiring an inﬁnite set of functions), the one-electron wave function could be exact solutions of the one-electron wave equation; in practise one selects a ﬁnite number of appropriate functions, generally

100 Quantum chemistry: solving the time-independent Schr¨ odinger equation

“Slater-type” functions that look like the 1s, 2s, 2p, ... hydrogen atom functions, which are themselves for computational reasons often composed of several local Gaussian functions. The one-electron wave functions are therefore approximations. The one-electron wave functions are ortho-normalized: (4.50) ψi |ψj = ψi∗ (r)ψj (r) dr = δij , and are completed to twice as many functions χ with spin α or β: χ2i−1 (x) = ψi (r)α(ω),

(4.51)

χ2i (x) = ψi (r)β(ω).

(4.52)

These χ-functions are also orthonormal, and are usually called Hartree–Fock spin orbitals. In order to construct the total wave function, ﬁrst the N -electron Hartree product function is formed: ΨHP = χi (x1 )χj (x2 ) . . . χk (xN ),

(4.53)

but this function does not satisfy the fermion parity rule. For example, for two electrons: ΨHP (x1 , x2 ) = χi (x1 )χj (x2 ) = −χj (x1 )χi (x2 ), while the following antisymmetrized function does: 1

Ψ(x1 , x2 ) = 2− 2 [χi (x1 )χj (x2 ) − χj (x1 )χi (x2 )] % % 1 %% χi (x1 ) χj (x1 ) %% . = √ % 2 χi (x2 ) χj (x2 ) %

(4.54)

In general, antisymmetrization is obtained by constructing the Slater determinant: % % % χi (x1 ) χj (x1 ) · · · χk (x1 ) % % % 1 %% χi (x2 ) χj (x2 ) · · · χk (x2 ) %% Ψ(x1 , x2 ), . . . , xN ) = √ % . (4.55) %. .. % N %% .. . % % χi (xN ) χj (xN ) · · · χk (xN ) % This has antisymmetric parity because any exchange of two rows (two particles) changes the sign of the determinant. The Slater determinant is abbreviated as Ψ = |χi χj . . . χk . Thus far we have not speciﬁed how the one-electron wave functions ψi ,

4.6 Hartree–Fock methods

101

and hence χi , are obtained. These functions are solutions of one-dimensional eigenvalue equations with a special Fock operator fˆ(i) instead of the hamiltonian: (4.56) fˆ(i)χ(r i ) = εχ(r i ) with 1 fˆ(i) = − ∇2i − 2

zk + v HF (i) rik

(4.57)

k

Here v HF is an eﬀective mean-ﬁeld potential that is obtained from the combined charge densities of all other electrons. Thus, in order to solve this equation, one needs an initial guess for the wave functions, and the whole procedure needs an iterative approach until the electronic density distribution is consistent with the potential v (self-consistent ﬁeld, SCF). For solving the eigenvalue equation (4.56) one applies the variational principle: for any function ψ = ψ0 , where ψ0 is the exact ground state solution ˆ = Eψ, the expectation value of H ˆ does not of the eigenvalue equation Hψ exceed the exact ground state eigenvalue E0 : ∗ ˆ dr ψ Hψ ≥ E0 (4.58) ψ ∗ ψ dr The wave function χ is varied (i.e., the coeﬃcients of its expansion in ba sis functions are varied) while keeping χ∗ χ dr = 1, until χ∗ f χ dr is a minimum. The electrons are distributed over the HF spin orbitals χ and form a conﬁguration. This distribution can be done by ﬁlling all orbitals from the bottom up with the available electrons, in which case a ground state conﬁguration is obtained. The energy of this “ground state” is called the Hartree–Fock energy, with the Hartree–Fock limit in the case that the basis set used approaches an inﬁnite set. But even the HF limit is not an accurate ground state energy because the whole SCF-HF procedure neglects the correlation energy between electrons. Electrons in the same spatial orbital (but obviously with diﬀerent spin state) tend to avoid each other and a proper description of the two-electron wave function should take the electron correlation into account, leading to a lower energy. This is also the case for electrons in diﬀerent orbitals. In fact, the London dispersion interaction between far-away electrons is based on electron correlation and will be entirely neglected in the HF approximation. The way out is to mix other, excited, conﬁgurations into the description of the wave function; in principle this conﬁguration interaction (CI) allows

102 Quantum chemistry: solving the time-independent Schr¨ odinger equation

for electron correlation. In practise the CI does not systematically converge and requires a huge amount of computational eﬀort. Modern developments use a perturbative approach to the electron correlation problem, such as the popular Møller–Plesset (MP) perturbation theory. For further details see Jensen (1999).

4.7 Density functional theory In SCF theory electron exchange is introduced through the awkward Slater determinant, while the introduction of electron correlation presents a major problem by itself. Density functional theory (DFT) oﬀers a radically diﬀerent approach that leads to a much more eﬃcient computational procedure. Unfortunately it is restricted to the ground state of the system. It has one disadvantage: the functional form needed to describe exchange and correlation cannot be derived from ﬁrst principles. In this sense DFT is not a pure ab initio method. Nevertheless: in its present form DFT reaches accuracies that can be approached by pure ab initio methods only with orders of magnitude higher computational eﬀort. In addition, DFT can handle much larger systems. The basic idea is that the electron charge density ρ(r) determines the exact ground state wave function and energy of a system of electrons. Although the inverse of this statement is trivially true, the truth of this statement is not obvious; in fact this statement is the ﬁrst theorem of Hohenberg and Kohn (1964). It can be rigorously proven. An intuitive explanation was once given by E. Bright Wilson at a conference in 1965:11 assume we know ρ(r). Then we see that ρ shows sharp maxima (cusps) at the positions of the nuclei. The local nuclear charge can be derived from the limit of the gradient of ρ near the nucleus, since at the nuclear position |∇ρ| = −2zρ. Thus, from the charge density we can infer the positions and charges of the nuclei. But if we know that and the number of electrons, the Hamiltonian is known and there will be a unique ground state solution to the time-independent Schr¨ odinger equation, specifying wave function and energy. Thus the energy and its constituent terms are functionals of the density ρ: E[ρ] = Vne [ρ] + K[ρ] + Vee [ρ] where

Vne =

11

(4.59)

ρ(r)vn (r) dr

Bright Wilson (1968), quoted by Handy (1996).

(4.60)

4.7 Density functional theory

103

is the electron–nuclear interaction, with vn the potential due to the nuclei, K is the kinetic energy of the electrons, and Vee is the electron–electron interaction which includes the mutual Coulomb interaction J: −1 ρ(r 1 )ρ(r 2 ), (4.61) J[ρ] = 12 dr 1 dr 2 r12 as well as the exchange and correlation contributions. Now, the second theorem of Hohenberg and Kohn states that for any density distribution ρ = ρ (where ρ is the exact ground state density), the energy is never smaller than the true ground state energy E: E[ρ ] ≥ E[ρ].

(4.62)

Thus ﬁnding ρ and E reduces to applying the variational principle to E[ρ], i.e., minimizing E by varying ρ(r), while keeping ρ(r) dr = N . Such a solution would provide the ground state energy and charge distribution, which is all we want to know: there is no need for knowledge of the detailed wave function. There is a slight problem, however: the functional form of the terms in (4.59) is not known! A practical solution was provided by Kohn and Sham (1965), who considered the equations that a hypothetical system of N non-interacting electrons should satisfy in order to yield the same density distribution as the real system of interacting electrons. Consider N non-interacting electrons in 12 N (+ 12 for odd N ) orbitals; the total properly antisymmetrized wave function would be the Slater determinant of the occupied spin-orbitals. For this system the exact expressions for the kinetic energy and the density are ni φ∗i (− 12 ∇2 )φi dr, (4.63) Ks [ρ] = i=1

ρ[r] =

ni φ∗i φi (r).

(4.64)

i=1

Here ni = 1 or 2 is the number of electrons occupying φi . The wave functions are solutions of the eigenvalue equation {− 12 ∇2 + vs (r)}φi = εi φi ,

(4.65)

where vs (r) is an as yet undetermined potential. The solution can be obtained by the variational principle, e.g., by expanding the functions φi in a suitable set of basis functions. Slater-type Gaussian basis sets may be used, but it is also possible to use a basis set of simple plane waves, particularly if the system under study is periodic.

104 Quantum chemistry: solving the time-independent Schr¨ odinger equation

In order to ﬁnd expressions for vs (r), we ﬁrst note that the energy functional of the non-interacting system is given by (4.66) E[ρ] = Ks [ρ] + vs (r)ρ(r) dr. The energy functional of the real interacting system is given by (4.59). Now writing the potential vs (r) in the hamiltonian for the Kohn-Sham orbitals (4.65) as ρ(r ) vs (r) = vn (r) + dr + vxc (r), (4.67) |r − r | the Kohn–Sham wave functions (or their expansion coeﬃcients in the chosen basis set) can be solved. In this potential the nuclear potential and the electrostatic electronic interactions are included; all other terms (due to electron correlation, exchange and the diﬀerence between the real kinetic energy and the kinetic energy of the non-interacting electrons) are absorbed in the exchange-correlation potential vxc . The equations must be solved iteratively until self-consistency because they contain the charge density that depends on the solution. Thus the Kohn–Sham equations are very similar to the SCF equations of Hartree–Fock theory. As long as no theory is available to derive the form of the exchangecorrelation potential from ﬁrst principles, approximations must be made. In its present implementation it is assumed that vxc depends only on local properties of the density, so that it will be expressible as a function of the local density and its lower derivatives. This excludes the London dispersion interaction, which is a correlation eﬀect due to dipole moments induced by distant ﬂuctuating dipole moments. The ﬁrst attempts to ﬁnd a form for the exchange-correlation functional (or potential) started from the exact result for a uniform electron gas, in which case the exchange potential is inversely proportional to the cubic root of the local density: 3 3 1/3 1/3 LDA =− ρ (4.68) vx 4 π so that the exchange functional Ex equals 3 3 1/3 LDA Ex [ρ] = − ρ4/3 (r) dr. 4 π

(4.69)

This local density approximation (LDA) is not accurate enough for atoms and molecules. More sophisticated corrections include at least the gradient of the density, as the popular exchange functional proposed by Becke (1988, 1992). With the addition of a proper correlation functional, as the Lee,

4.8 Excited-state quantum mechanics

105

Yang and Parr functional (Lee et al., 1988), which includes both ﬁrst and second derivatives of the density, excellent accuracy can be obtained for structures and energies of molecules. The combination of these exchange and correlation functionals is named the BLYP exchange-correlation functional. A further modiﬁcation B3LYP exists (Becke, 1993). The functional forms can be found in Leach (2001).

4.8 Excited-state quantum mechanics Normally, quantum-chemical methods produce energies and wave functions (or electron densities) for the electronic ground state. In many applications excited-state properties are required. For the prediction of spectroscopic properties one wishes to obtain energies of selected excited states and transition moments between the ground state and selected excited states. For the purpose of simulation of systems in which excited states occur, as in predicting the fate of optically excited molecules, one wishes to describe the potential energy surface of selected excited states, i.e., the electronic energy as a function of the nuclear coordinates. While dynamic processes involving electronically excited states often violate the Born–Oppenheimer approximation and require quantum-dynamical methods, the latter will make use of the potential surfaces of both ground and excited states, generated under the assumption of stationarity of the external potential (nuclear positions). Within the class of Hartree–Fock methods, certain excited states, deﬁned by the conﬁguration of occupied molecular orbitals, can be selected and optimized. In the conﬁguration interaction (CI) scheme to incorporate electron correlation, such excited states are considered, and used to mix with the ground state. The popular complete active space SCF (CASSCF) method of Roos (1980) can also be applied to speciﬁc excited-state conﬁgurations and produce excited-state potential surfaces. Unfortunately, density-functional methods are only valid for the ground state and cannot be extended to include excited states. However, not all is lost, as time-dependent DFT allows the prediction of excited-state properties. The linear response of a system to a periodic perturbation (e.g., an electric ﬁeld) can be computed by DFT; excited states show up by a peak in absorbance, so that at least their relative energies and transition moments can be computed. If this is done for many nuclear conﬁgurations, the excited-state energy surface can be probed. This application is not straightforward, and, thus far, DFT has not been used much for the purpose of generating excited-state energy surfaces.

106 Quantum chemistry: solving the time-independent Schr¨ odinger equation

4.9 Approximate quantum methods While DFT scales more favorably with system size than extended HartreeFock methods, both approaches are limited to relatively small system sizes. This is particularly true if the electronic calculation must be repeated for many nuclear conﬁgurations, such as in molecular dynamics applications. In order to speed up the electronic calculation, many approximations have been proposed and implemented in widespread programs. Approximations to HF methods involve: (i) restricting the quantum treatment to valence electrons; (ii) restricting the shape of the atomic orbitals, generally to Slater-type orbitals (STO) of the form rn−1 exp(−αr)Ylm (θ, φ); (iii) neglecting or simplifying the overlap between neighboring atomic orbitals; (iv) neglecting small integrals that occur in the evaluation of the Hamiltonian needed to minimize the expectation of the energy (4.58); (v) replacing other such integrals by parameters. Such methods require parametrization based on experimental (structural, thermodynamic and spectroscopic) data and are therefore classiﬁed as semiempirical methods. This is not the place to elaborate on these methods; for a review the reader is referred to Chapter 5 of Cramer (2004). Suﬃce to say that of the numerous diﬀerent approximations, the MNDO (modiﬁed neglect of diﬀerential overlap) and NDDO (neglect of diatomic diﬀerential overlap) methods seem to have survived. Examples of popular approaches are AM1 (Dewar et al., 1985: the Austin Model 1) and the better parameterized PM3 (Stewart, 1989a, 1989b: Parameterized Model 3), which are among others available in Stewart’s public domain program MOPAC 7. Even semi-empirical methods do not scale linearly with the number of atoms in the system, and are not feasible for systems containing thousands of atoms. For such systems one looks for linear-scaling methods, such as the DAC (“divide-and-conquer”) DFT scheme of Yang (1991a, 1991b). In this scheme the system is partitioned into local areas (groups of atoms, or even atoms themselves), where the local density is computed directly from a density functional, without evoking Kohn–Sham orbitals. One needs a local Hamiltonian which is a projection of the Hamiltonian onto the local partition. The local electron occupation is governed by a global Fermi level (electron free energy), which is determined by the total number of electrons in the system. This description has been improved by a formulation in terms of local density matrices (Yang and Lee, 1995) and promises to be applicable to very large molecules (Lee et al., 1996).

4.10 Nuclear quantum states

107

For solids, an empirical approach to consider the wave function as a linear combination of atomic orbitals with ﬁtted parameters for the interactions and overlap, is known under the name tight-binding approximation. The TB approximation is suitable to be combined with molecular dynamics (Laasonen and Nieminen, 1990). 4.10 Nuclear quantum states While this chapter has so far dealt only with electronic states in stationary environments, nuclear motion, if undisturbed and considered over long periods of time, will also develop into stationary states, governed by the time-independent Schr¨ odinger equation. The knowledge of such nuclear rotational-vibrational states is useful in connection with infrared and Raman spectroscopy. We shall assume the Born–Oppenheimer approximation (discussed in Section 4.4) to be valid, i.e., for each nuclear conﬁguration the electronic states are pure solutions of the time-independent Schr¨ odinger equation, as if the nuclei do not move, and thus the electrons provide a potential ﬁeld for the nuclear interactions. The electrons have been factored-out of the complete nuclear-electronic wave function, and the electronic degrees of freedom do not occur in the nuclear Schr¨ odinger equation

2 2 − ∇ + V (r 1 , . . . r N ) Ψ = EΨ, (4.70) 2mi i i

where i = 1, . . . , N enumerates the nuclei, mi is the nuclear mass, Ψ is a function of the nuclear coordinates and V is the interaction potential function of the nuclei, including the inﬂuence of the electrons. For every electronic state there is a diﬀerent potential function and a diﬀerent set of solutions. The computation of eigenstates (energies and wave functions) is in principle not diﬀerent from electronic calculations. Since there is always a strong repulsion at small distances between nuclei in molecules, exchange can be safely neglected. This considerably reduces the complexity of the solution. This also implies that the spin states of the nuclei generally have no inﬂuence on the energies and spatial wave functions of the nuclear eigenstates. However, the total nuclear wave function of a molecule containing identical nuclei must obey the symmetry properties of bosons or fermions (whichever is applicable) when two identical nuclei are exchanged. This leads to symmetry requirements implying that certain nuclear states are not allowed. For isolated molecules or small molecular complexes, the translational motion factors out, but the rotational and vibrational modes couple into

108 Quantum chemistry: solving the time-independent Schr¨ odinger equation

vibrational-rotational-tunneling (VRT) states. While the energies and wave functions for a one-dimensional oscillator can be computed easily by numerical methods, as treated in Section 4.3, in the multidimensional case the solution is expressed in a suitable set of basis functions. These are most easily expressed as functions of internal coordinates, like Euler angles and intramolecular distances and angles, taking symmetry properties into account. The use of internal coordinates implies that the kinetic energy operator must also be expressed in internal coordinates, which is not entirely trivial. We note that the much more easily obtained classical solution of internal vibrational modes corresponds to the quantum solution only in the case that rotational and vibrational modes are separable, and the vibration is purely harmonic. The complete treatment of the VRT states for molecular complexes is beyond the scope of this book. The reader is referred to an excellent review by Wormer and van der Avoird (2000) describing the methods to compute VRT states in van der Waals complexes like argon-molecule clusters and hydrogenbonded complexes like water clusters. Such weakly bonded complexes often havemultiple minima connected through relatively low saddle-point regions, thus allowing for eﬀective tunneling between minima. The case of the water dimer, for which highly accurate low-frequency spectroscopic data are available, both for D2 O (Braly et al., 2000a) and H2 O (Braly et al., 2000b), has received special attention. There are eight equivalent global minima, all connected by tunneling pathways, in a six-dimensional intermolecular vibration-rotation space (Leforestier et al., 1997; Fellers et al., 1999). The comparison of predicted spectra with experiment provides an extremely sensitive test for intermolecular potentials.

5 Dynamics of mixed quantum/classical systems

5.1 Introduction We now move to considering the dynamics of a system of nuclei and electrons. Of course, both electrons and nuclei are subject to the laws of quantum mechanics, but since nuclei are 2000 to 200 000 times heavier than electrons, we expect that classical mechanics will be a much better approximation for the motion of nuclei than for the motion of electrons. This means that we expect a level of approximation to be valid, where some of the degrees of freedom (d.o.f.) of a system behave essentially classically and others behave essentially quantum-mechanically. The system then is of a mixed quantum/classical nature. Most often the quantum subsystem consists of system of electrons in a dynamical ﬁeld of classical nuclei, but the quantum subsystem may also be a selection of generalized nuclear coordinates (e.g., corresponding to highfrequency vibrations) while other generalized coordinates are supposed to behave classically, or describe the motion of a proton in a classical environment. So, in this chapter we consider the dynamics of a quantum system in a non-stationary potential. In Section 5.2 we consider the time-dependent potential as externally given, without taking notice of the fact that the sources of the time-dependent potential are moving nuclei, which are quantum particles themselves, feeling the interaction with the quantum d.o.f. Thus we consider the time evolution of the quantum system, which now involves mixing-in of excited states, but we completely ignore the back reaction of the quantum system onto the d.o.f. that cause the time-dependent potential, i.e., the moving nuclei. In this way we avoid the main diﬃculty of mixing quantum and classical d.o.f.: how to reconcile the evolution of probability density of the quantum particles with the evolution of the trajectory 109

110

Dynamics of mixed quantum/classical systems

of the classical particles. The treatment in this approximation is applicable to some practical cases, notably when the energy exchange between classical and quantum part is negligible (this is the case, for example, for the motion of nuclear spins in a bath of classical particles at normal temperatures), but will fail completely when energy changes in the quantum system due to the external force are of the same order as the energy ﬂuctuations in the classical system. How the quantum subsystem can be properly embedded in the environment, including the back reaction, is considered in Section 5.3. As we shall see in Section 5.2, the eﬀect of time-dependent potentials is that initially pure quantum states evolve into mixtures of diﬀerent states. For example, excited states will mix in with the ground state as a result of a time-dependent potential. Such time dependence may arise from a time-dependent external ﬁeld, as a radiation ﬁeld that causes the system to “jump” to an excited state. It may also arise from internal interactions, such as the velocity of nuclei that determine the potential ﬁeld for the quantum system under consideration, or from thermal ﬂuctuations in the environment, as dipole ﬂuctuations that cause a time-dependent electric ﬁeld. The wave functions that result do not only represent additive mixtures of different quantum states, but the wave function also carries information on phase coherence between the contributing states. The mixed states will in turn relax under the inﬂuence of thermal ﬂuctuations that cause dephasing of the mixed states. The occurrence of coherent mixed states is a typical quantum behavior, for which there is no classical analog. It is the cause of the diﬃculty to combine quantum and classical treatments, and of the diﬃculty to properly treat the back reaction to the classical system. The reason is that the wave function of a dephased mixed state can be viewed as the superposition of diﬀerent quantum states, each with a given population. Thus the wave function does not describe one trajectory, but rather a probability distribution of several trajectories, each with its own back reaction to the classical part of the system. If the evolution is not split into several trajectories, and the back reaction is computed as resulting from the mixed state, one speaks of a mean-ﬁeld solution, which is only an approximation. When the quantum system is in the ground state, and all excited states have energies so high above the ground state that the motions of the “classical” degrees of freedom in the system will not cause any admixture of excited states, the system remains continuously in its ground state. The evolution now is adiabatic, as there is no transfer of “heat” between the classical environment and the quantum subsystem. In that case the back reaction is simply the expectation of the force over the ground-state wave function, and a consistent mixed quantum-classical dynamics is obtained.

5.1 Introduction

6

@ @ @ A @ @ @

B

@ @ @ @ @

energy

B

-

@ @ A @ @ @

111

@ @ @

A

@ 6 6 @ @ - ΔE0 @ splitting 2C @ @ ? ? @ @ B @ @

nuclear coordinate

Figure 5.1 Two crossing diabatic states A and B. Precise solution in the crossing region (right) yields two adiabatic non-crossing states A and B . Time dependence may cause transitions in the crossing region.

One method in this category, the Car–Parrinello method, also referred to as ab initio molecular dynamics (see Section 6.3.1), has proved to be very successful for chemically reactive systems in the condensed phase. In cases where excited states are relevant, adiabatic dynamics is not sufﬁcient and the separation between quantum and classical d.o.f. is no longer trivial. Now we are fully confronted with the question how to treat the evolution into a multitude of trajectories and how to evaluate the back reaction of the quantum system onto the classical d.o.f. Consider the case that the quantum system develops into a mixture of two “pure” states. This could easily happen if the trajectory arrives at a point where the quantum system is degenerate or almost degenerate, i.e., where two states of the quantum system cross or nearly cross (see Fig. 5.1). When there is a small coupling term H12 = H21 = C between the two states, the hamiltonian in the neighborhood of the crossing point will be: 1 C − 2 ΔE0 , (5.1) H= 1 C 2 ΔE0 and the wave functions will mix. The eigenvalues are & 1 E1,2 = ± (ΔE0 )2 + C 2 . 4

(5.2)

112

Dynamics of mixed quantum/classical systems

At the crossing point (ΔE0 = 0), the adiabatic solutions are equal mixtures of both diabatic states, with a splitting of 2C. Then there will be essentially two trajectories of the classical system possible, each related to one of the pure states. The system “choses” to evolve in either of the two branches. Only by taking the quantum character of the “classical” system into account can we fully understand the behavior of the system as a whole; the full wave function of the complete system would describe the evolution. Only that full wave function will contain the branching evolution into two states with the probabilities of each. In that full wave function the two states would still be related to each other in the sense that the wave functions corresponding to the two branches are not entirely separated; their phases remain correlated. In other words, a certain degree of “coherence” remains also after the “splitting” event, until random external disturbances destroy the phase relations, and the two states can be considered as unrelated. The coherence is related to reversibility: as long as the coherence is not destroyed, the system is time-reversible and will retrace its path if it returns to the same conﬁguration (of the “classical” d.o.f.) where the splitting originally occurred. Such retracing may occur in small isolated systems (e.g., a diatomic molecule in the gas phase) if there is a reﬂection or a turning point for the classical d.o.f., as with the potential depicted in Fig. 5.2; in many-particle systems such revisiting of previously visited conﬁgurations becomes very unlikely. If in the mean time the coherence has been destroyed, the system has lost memory of the details of the earlier splitting event and will not retrace to its original starting point, but develop a new splitting event on its way back. If we would construct only one trajectory based on the expectation value of the force, the force would be averaged over the two branches, and – assuming symmetry and equal probabilities for both branches (Fig. 5.1) after the splitting – the classical d.o.f. would feel no force and proceed with constant velocity. In reality the system develops in either branch A, continuously accelerating, or branch B, decelerating until the turning point is reached. It does not do both at the same time. Thus, the behavior based on the average force, also called the mean-ﬁeld treatment, is clearly incorrect. It will be correct if the system stays away from regions where trajectories may split up into diﬀerent branches, but cannot be expected to be correct if branching occurs. In Section 5.3 simulations in a mixed quantum-classical system with back reaction are considered. The simplest case is the mean-ﬁeld approach (Section 5.3.1), which gives consistent dynamics with proper conservation of energy and momentum over the whole system. However, it is expected to be valid only for those cases where either the back reaction does not notice-

5.1 Introduction

113

E

A 6

B q

U

G L

nuclear coordinate Figure 5.2 Two crossing diabatic states G and E, developing into two adiabatic states U (upper) and L (lower). After excitation from G to E (reaching point A) the system either stays on the adiabatic level U or crosses diabatically to L, depending on coupling to dynamical variables in the crossing region. If it stays in U, it reaches a turning point B and retraces its steps, if in the meantime no dephasing has taken place due to external ﬂuctuations. With dephasing, the system may cross to G on the way back and end up vibrating in the ground state.

ably inﬂuence the classical system, or the nuclei remain localized without branching. An approximation that catches the main deﬁciency of the meanﬁeld treatment is the surface-hopping procedure (Section 5.3.3), introduced by Tully (1990). The method introduces random hops between diﬀerent potential energy surfaces, with probabilities governed by the wave function evolution. So the system can also evolve on excited-state surfaces, and incorporate non-adiabatic behavior. Apart from the rather ad hoc nature of the surface-hopping method, it is incorrect in the sense that a single, be it stochastic, trajectory is generated and the coherence between the wave functions on diﬀerent trajectories is lost. Attempts to remedy this deﬁciency and to validate the surface-hopping method include multitrajectory methods that preserve the coherence between trajectories. Although not often applied, the mixed quantum-classical system could also be treated using the Bohmian particle description of quantum mechanics, introduced in Section 3.4. A Bohmian trajectory is a sample from the possible evolutions, and the selection of a particular trajectory allows to handle the back reaction

114

Dynamics of mixed quantum/classical systems

for that particular trajectory (Section 5.3.4). In order to be able to handle the full back reaction, an ensemble of Bohmian trajectories will be needed. The mixed quantum-classical behavior is not limited to electrons and nuclei; we can just as well treat the quantum behavior of selected nuclei (notably the proton) in a dynamical classical environment. In this case the electrons are treated in the full Born–Oppenheimer approximation; they are consistently in the electronic ground state and provide a potential ﬁeld for the nuclear interactions. But the quantum nucleus has the ability to tunnel between energy wells, and move non-adiabatically with the involvement of nuclear excited states. It should be noted that the quantum eﬀects of nuclei, being so much closer to classical behavior than electrons, can often be treated by the use of eﬀective potentials in classical simulations, or even by quantum corrections to classical behavior. Section 3.5 is devoted to these approximate methods.

5.2 Quantum dynamics in a non-stationary potential Assume that a quantum system of n particles r 1 , . . .r n interacts with a time-dependent Hamiltonian ˆ ˆ + Vˆ (t). H(t) =K

(5.3)

Then the time-dependent Schr¨ odinger equation i ˆ ∂ Ψ(r, t) = − H(t)Ψ(r, t) (5.4) ∂t can formally be solved as (see Chapter 14 for details of exponential operators) i t ˆ (5.5) H(t ) dt Ψ(r, 0). Ψ(r, t) = exp − 0 This way of writing helps us to be concise, but does not help very much to actually solve the time-dependent wave function. In most applications a time-dependent solution in terms of expansion on a basis set is more suitable than a direct consideration of the wave function itself. If the time-dependence is weak, i.e., if the time-dependent interaction can be considered as a perturbation, or if the time dependence arises from the parametric dependence on slowly varying nuclear coordinates, the solution approaches a steady-state. Using expansion in a set of eigenfunctions of the time-independent Schr¨ odinger equation, all the hard work is then done separately, and the time-dependence is expressed as the way the timeindependent solutions mix as a result of the time-dependent perturbation.

5.2 Quantum dynamics in a non-stationary potential

115

The basis functions in which the time-dependent wave function is expanded can be either stationary or time-dependent, depending on the character of the problem. For example, if we are interested in the action of timedependent small perturbations, resulting from ﬂuctuating external ﬁelds, on a system that is itself stationary, the basis functions will be stationary as well. On the other hand, if the basis functions are solutions of the time-independent Schr¨ odinger equation that contains moving parameters (as nuclear positions), the basis functions are themselves time-dependent. The equations for the mixing coeﬃcients are diﬀerent in the two cases. In the following subsections we shall ﬁrst consider direct evolution of the wave function in a time-dependent ﬁeld by integration on a grid (Section 5.2.1), followed by a consideration of the evolution of the vector representing the wave function in Hilbert space. In the latter case we distinguish two cases: the basis set itself is either time-independent (Section 5.2.2) or time-dependent (Section 5.2.3). 5.2.1 Integration on a spatial grid One way to obtain a solution is to describe the wave function on a spatial grid, and integrate values on the grid in small time steps. This actually works well for a single particle in up to three dimensions, where a grid of up to 1283 can be used, and is even applicable for higher dimensionalities up to six, but is not suitable if the quantum system contains several particles. For the actual solution the wave function must be sampled on a grid. Consider for simplicity the one-dimensional case, with grid points x0 = 0, x1 = δ, . . . , xn = nδ, . . . xL = Lδ. We assume that boundary conditions for the wave function are given (either periodic, absorbing or reﬂecting) and that the initial wave function Ψ(x, t) is given as well. We wish to solve ∂Ψ(x, t) i 2 ∂ 2 =− − + V (x, t) Ψ. (5.6) ∂t 2m ∂x2 The simplest discretization of the second spatial derivative1 is ∂2Ψ Ψn−1 − 2Ψn + Ψn+1 = , 2 ∂x δ2 yielding ∂Ψ(x, t) i = − ∂t 1

(5.7)

2 2 + Vn Ψn − (Ψn−1 + Ψn+1 ) mδ 2 2mδ 2

The error in this three-point discretization is of order δ 2 . A ﬁve-point discretization with error of order δ 4 is (1/12δ 2 )(−Ψn−2 + 16Ψn−1 − 30Ψn + 16Ψn+1 − Ψn+2 ). See Abramowitz and Stegun (1965) for more discretizations of partial derivatives.

116

Dynamics of mixed quantum/classical systems

= −i(HΨ)n , where Vn = V (xn , t) and ⎛ a0 ⎜ b ⎜ ⎜ ⎜ ⎜ H=⎜ ⎜ ⎜ ⎜ ⎜ ⎝

b a1 .. .

(5.8) ⎞ b .. b

.

..

.

an .. .

b .. b

.

..

.

aL−1 b

⎟ ⎟ ⎟ ⎟ ⎟ ⎟. ⎟ ⎟ ⎟ ⎟ b ⎠ aL

Note that we have absorbed into H. The matrix elements are Vn , b=− + . an = 2 mδ 2mδ 2 We seek the propagation after one time step τ : Ψ(t + τ ) = e−iHτ Ψ(t),

(5.9)

(5.10)

(5.11)

using an approximation that preserves the unitary character of the propagator in order to obtain long-term stability of the algorithm. A popular unitary approximation is the Crank–Nicholson scheme:2 UCN = (1 + iHτ /2)−1 (1 − iHτ /2),

(5.12)

which is equivalent to (1 + iHτ /2)Ψ(t + τ ) = (1 − iHτ /2)Ψ(t).

(5.13)

Both sides of this equation approximate Ψ(t+τ /2), the left side by stepping τ /2 backward from Ψ(t+τ ) and the right side by stepping τ /2 forward from Ψ(t + τ ) (see Fig. 5.3). The evaluation of the ﬁrst line requires knowledge of Ψ(t + τ ), which makes the Crank–Nicholson step into an implicit scheme: [2 + iτ an (t + τ )]Ψn (t + τ ) − ibτ [Ψn−1 (t + τ ) + Ψn+1 (t + τ )] = [2 − iτ an (t)]Ψn (t) − ibτ [Ψn−1 (t) + Ψn+1 (t)].

(5.14)

Finding Ψ(t + τ ) requires the solution of a system of equations involving a tridiagonal matrix, which can be quickly solved. This, however is only true for one dimension; for many dimensions the matrix is no longer tridiagonal and special sparse-matrix techniques should be used. An elegant and eﬃcient way to solve (5.11) has been devised by de Raedt 2

Most textbooks on partial diﬀerential equations will treat the various schemes to solve initial value problems like this one. In the context of the Schr¨ odinger equation Press et al. (1992) and Vesely (2001) are useful references.

5.2 Quantum dynamics in a non-stationary potential

117

time ↑ t+τ t+τ/2 t n-1

n

→x/δ

n+1

Figure 5.3 Space-time grid for the implicit, time-reversible Crank–Nicholson scheme. A virtual point at xn , t + τ /2 is approached from t and from t + τ , yielding U (−τ /2)Ψ(t + τ ) = U (τ /2)Ψ(t).

(1987, 1996), using the split-operator technique (see Chapter 14). It is possible to split the operator H in (5.9) into a sum of easily and exactly solvable block-diagonal matrices, such as H = H1 + H2 , with

⎞

⎛

b a0 ⎜ b a1 /2 ⎜ ⎜ a2 /2 b H1 = ⎜ ⎜ b a 3 /2 ⎝ ⎛ ⎜ ⎜ ⎜ ⎜ H2 = ⎜ ⎜ ⎜ ⎝

..

⎟ ⎟ ⎟ ⎟ ⎟ ⎠

(5.15)

. ⎞

0

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

b a1 /2 b a2 /2 b a3 /2 b a4 /2 ..

(5.16)

.

Each 2 × 2 exponential matrix can be solved exactly, and independently of other blocks, by diagonalization (see Chapter 14, page 388), yielding a 2 × 2 matrix. Then the total exponential matrix can be applied using a form of Trotter–Suzuki splitting, e.g., into the second-order product (see

118

Dynamics of mixed quantum/classical systems

Chapter 14, page 386) e−iHτ = e−iH1 τ /2 e−iH2 τ e−iH1 τ /2 .

(5.17)

The Hamiltonian matrix can also be split up into one diagonal and two block matrices of the form cos bτ −i sin bτ 0 b , (5.18) = exp −iτ −i sin bτ cos bτ b 0 simplifying the solution of the exponential block matrices even more. For d dimensions, the operator may be split into a sequence of operators in each dimension.3 While the methods mentioned above are pure real-space solutions, one ˆ and may also split the Hamiltonian operator into the kinetic energy part K ˆ the potential energy part V . The latter is diagonal in real space and poses no problem. The kinetic energy operator, however, is diagonal in reciprocal space, obtained after Fourier transformation of the wave function. Thus the exponential operator containing the kinetic energy operator can be applied easily in Fourier space, and the real-space result is then obtained after an inverse Fourier transformation. An early example of the evolution of a wave function on a grid, using this particular kind of splitting, is the study of Selloni et al. (1987) on the evolution of a solvated electron in a molten salt (KCl). The electron occupies local vacancies where a chloride ion is missing, similar to F-centers in solid salts, but in a very irregular and mobile way. The technique they use relies on the Trotter expansion of the exponential operator and uses repeated Fourier transforms between real and reciprocal space. For a small time step Δt the update in Ψ is approximated by i t+Δt ˆ ˆ Ψ(t + Δt) = exp − [K + V (t )] dt Ψ(r, t) t ˆ iVˆ (t + 12 Δt)Δt iKΔt exp − ≈ exp − 2 ˆ iKΔt Ψ(r, t). (5.19) × exp − 2 Here we have used in the last equation the Trotter split-operator approximation for exponential operators with a sum of non-commuting exponents 3

De Raedt (1987, 1996) employs an elegant notation using creation and annihilation operators to index oﬀ-diagonal matrix elements, thus highlighting the correspondence with particle motion on lattice sites, but for the reader unfamiliar with Fermion operator algebra this elegance is of little help.

5.2 Quantum dynamics in a non-stationary potential

119

(see Chapter 14), which is of higher accuracy than the product of two exponential operators. Note that it does not matter what the origin of the time-dependence of V actually is: V may depend parametrically on timedependent coordinates of classical particles, it may contain interactions with time-dependent ﬁelds (e.g., electromagnetic radiation) or it may contain stochastic terms due to external ﬂuctuations. The operator containing V is straightforwardly applied to the spatial wave function, as it corresponds simply to multiplying each grid value with a factor given by the potential, ˆ involves a second derivative over the grid. but the operator containing K ˆ is simply proThe updating will be straightforward in reciprocal space, as K portional to k 2 . Using fast Fourier transforms (FFT), the wave function can ˆ applied, ﬁrst be transformed to reciprocal space, then the operator with K and ﬁnally transformed back into real space to accomplish the evolution in the kinetic energy operator. When the motion does not produce ﬂuctuations in the potential in the frequency range corresponding to transition to the excited state, i.e., if the energy gap between ground and excited states is large compared to kB T , the solvated electron behaves adiabatically and remains in the ground Born–Oppenheimer state.

5.2.2 Time-independent basis set Let us ﬁrst consider stationary basis functions and a Hamiltonian that contains a time-dependent perturbation: ˆ 1 (t). ˆ =H ˆ0 + H H

(5.20)

ˆ 0: Assume that the basis functions φn (r) are solutions of H ˆ 0 φn = En φn , H

(5.21)

so that the Hamiltonian matrix H0 is diagonal on that basis set (see Chapter 14). We write the solution of the time-dependent equation i ˆ0 ∂ ˆ 1 (t))Ψ Ψ(r, t) = − (H +H ∂t as a linear combination of time-independent basis functions: cn (t)φn (r). Ψ(r, t) =

(5.22)

(5.23)

n

Note that cn (t) contains an oscillating term exp(−iωn t), where ωn = En /.

120

Dynamics of mixed quantum/classical systems

The time-dependent Schr¨ odinger equation now implies i ∂ ˆ m, cm φm = c˙m φm = − cm Hφ ∂t m m

(5.24)

which, after left-multiplying with φn and integrating, results in c˙n = −

i ˆ m = − i (Hc)n , cm φn |H|φ m

(5.25)

or in matrix notation: i c˙ = − Hc.

(5.26)

Since H is diagonal, there are two terms in the time-dependence of cn : i c˙n = −iωn cn − (H1 c)n .

(5.27)

The ﬁrst term simply gives the oscillating behavior exp(−iωn t) of the unperturbed wave function; the second term describes the mixing-in of other states due to the time-dependent perturbation. Often a description in terms of the density matrix ρ is more convenient, as it allows ensemble-averaging without loss of any information on the quantum behavior of the system (see Chapter 14). The density matrix is deﬁned by ρnm = cn c∗m

(5.28)

and its equation of motion on a stationary basis set is the Liouville-Von Neumann equation: ρ˙ =

i i [ρ, H] = [ρ, H0 + H1 (t)]

(5.29)

The diagonal terms of ρ, which do not oscillate in time, indicate how much each state contributes to the total wave function probability Ψ∗ Ψ and can be interpreted as a population of a certain state; the oﬀ-diagonal terms ρnm , which oscillate because they contain a term exp(−i(ωn − ωm )t), reveal a coherence in the phase behavior resulting from a speciﬁc history (e.g., a recent excitation). If averaged over an equilibrium ensemble, the oﬀ-diagonal elements cancel because their phases are randomly distributed. See also Section 14.8. In Section 5.2.4 of this chapter it will be shown for a two-level system how a randomly ﬂuctuating perturbation will cause relaxation of the density matrix.

5.2 Quantum dynamics in a non-stationary potential

121

5.2.3 Time-dependent basis set We now consider the case that the basis functions are time-dependent themselves through the parametric dependence on nuclear coordinates R: φn = φn (r; R(t)). They are eigenfunctions of the time-independent Schr¨ odinger equation, in which the time dependence of R is neglected. The total wave function is expanded in this basis set: cn φn . (5.30) Ψ= n

Inserting this into the time-dependent Schr¨ odinger equation (5.22) we ﬁnd (see also Section 14.7 and (14.54)) i ˙ · ˆ m. c˙m φm + R cm ∇R φm = − cm Hφ (5.31) m m m After left-multiplying with φ∗n and integrating, the equation of motion for the coeﬃcient cn is obtained: i ˙ · cm φn |∇R |φm = − (Hc)n , (5.32) c˙n + R m which in matrix notation reads i ˙ · D)c. c˙ = − (H + R (5.33) Here D is the matrix representation of the non-adiabatic coupling vector ˆ = −i∇ ˆR : operator D (5.34) Dnm = φn |∇R |φm . i Note that D is purely imaginary. D is a vector in the multidimensional space of the relevant nuclear coordinates R, and each vector element is an operator (or matrix) in the Hilbert space of the basis functions. In terms of the density matrix, the equation of motion for ρ for nonstationary basis functions then is i ˙ · D]. [ρ, H + R (5.35) The shape of this equation is the same as in the case of time-independent basis functions (5.29): the Hamiltonian is modiﬁed by a (small) term, which can often be treated as a perturbation. The diﬀerence is that in (5.29) the perturbation comes from a, usually real, term in the potential energy, while in (5.35) the perturbation is imaginary and proportional to the velocity of the sources of the potential energy. Both perturbations may be present ρ˙ =

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simultaneously. We note that in the literature (see, e.g., Tully, 1990) the real and antisymmetric (but non-Hermitian) matrix element (which is a vector in R-space) def

dnm = φn |∇R |φm =

i D nm

(5.36)

is often called the non-adiabatic coupling vector ; its use leads to the following equation of motion for the density matrix: ρ˙ =

i ˙ [ρ, H] + R[ρ, d].

(5.37)

In the proof of (5.35), given below, we make use of the fact that D is a Hermitian matrix (or operator): D† = D.

(5.38)

ˆ cannot change the value of φn |φm , which This follows from the fact that D equals δnm and is thus independent of R: ∗ ˆ (5.39) D φn φm dr = 0. Hence it follows that

ˆ ∗n )φm dr + (Dφ

ˆ m dr = 0 φ∗n Dφ

or, using the fact that D∗ = −D, ∗ ∗ ˆ ˆ m dr = 0, − φm Dφn dr + φ∗n Dφ

(5.40)

(5.41)

meaning that D† = D.

(5.42)

Since D is imaginary, Dnm = −Dmn and Dnn = 0. ˙ † + cc˙ † , we Proof We prove (5.35). Realizing that ρ = cc† and hence ρ˙ = cc ﬁnd that i ˙ · D)ρ + i ρ(H† + R ˙ · D† ) ρ˙ = − (H + R . Using the fact that both H and D are Hermitian, (5.35) follows.

5.2 Quantum dynamics in a non-stationary potential

123

5.2.4 The two-level system It is instructive to consider a quantum system with only two levels. The extension to many levels is quite straightforward. Even if our real system has multiple levels, the interesting non-adiabatic events that take place under the inﬂuence of external perturbations, such as tunneling, switching from one state to another or relaxation, are active between two states that lie close together in energy. The total range of real events can generally be built up from events between two levels. We shall use the density matrix formalism (see Section 14.8),4 as this leads to concise notation and is very suitable for extension to ensemble averages. We start with a description based on a diagonal zero-order Hamiltonian H0 plus a time-dependent perturbation H1 (t). The perturbation may arise from interactions with the environment, as externally applied ﬁelds or ﬂuctuating ﬁelds from thermal ﬂuctuations, but may also arise from motions of the nuclei that provide the potential ﬁeld for electronic states, as described in the previous section. The basis functions are two eigenfunctions of H0 , with energies E10 and E20 . The equation of motion for the density matrix is (5.29): ρ˙ =

i i [ρ, H] = [ρ, H0 + H1 (t)].

(5.43)

Since tr ρ = 1, it is convenient to deﬁne a variable: z = ρ11 − ρ22 ,

(5.44)

instead of ρ11 and ρ22 . The variable z indicates the population diﬀerence between the two states: z = 1 if the system is completely in state 1 and z = −1 if it is in state 2. We then have the complex variable ρ12 and the real variable z obeying the equations: i [ρ12 (H22 − H11 ) + zH12 ], 2i ∗ − ρ∗12 H12 ], z˙ = [ρ12 H12

ρ˙ 12 =

(5.45) (5.46)

where we have used the Hermitian property of ρ and H. Equations (5.45) and (5.46) imply that the quantity 4ρ12 ρ∗12 + z 2 is a constant of the motion since the time derivative of that quantity vanishes. Thus, if we deﬁne a real 4

See Berendsen and Mavri (1993) for density-matrix evolution (DME) in a two-level system; the theory was applied to proton transfer in aqueous hydrogen malonate by Mavri et al. (1993). The proton transfer case was further extended to multiple states by Mavri and Berendsen (1995) and summarized by Berendsen and Mavri (1997). Another application is the perturbation of a quantum oscillator by collisions, as in diatomic liquids (Mavri et al., 1994).

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Dynamics of mixed quantum/classical systems

three-dimensional vector r with components x, y, z, with: x = ρ12 + ρ∗12 , y = −i(ρ12 −

ρ∗12 ),

(5.47) (5.48)

then the length of that vector is a constant of the motion. The motion of r is restricted to the surface of a sphere with unit radius. The time-dependent perturbation causes this vector to wander over the unit sphere. The equation of motion for r(t) can now be conveniently expressed in the perturbations when we write the latter as three real time-dependent angular frequencies: 1 ∗ (H12 + H12 ), def 1 ∗ (H12 − H12 ), ωy (t) = i def 1 ∗ ), ωz (t) = (H11 − H22

(5.50)

x˙ = yωz − zωy ,

(5.52)

y˙ = zωx − xωz ,

(5.53)

z˙ = xωy − yωx .

(5.54)

def

ωx (t) =

(5.49)

(5.51)

yielding

These equations can be summarized as one vector equation: r˙ = r × ω.

(5.55)

Equation (5.55) describes a rotating top under the inﬂuence of a torque. This equivalence is in fact well-known in the quantum dynamics of a two-spin system (Ernst et al., 1987), where r represents the magnetization, perturbed by ﬂuctuating local magnetic ﬁelds. It gives some insight into the relaxation behavior due to ﬂuctuating perturbations. The oﬀ-diagonal perturbations ωx and ωy rotate the vector r in a vertical plane, causing an oscillatory motion between the two states when the perturbation is stationary, but a relaxation towards equal populations when the perturbation is stochastic.5 In other words, oﬀ-diagonal perturbations cause transitions between states and thus limit the lifetime of each state. In the language of spin dynamics, oﬀ-diagonal stochastic perturbations cause longitudinal relaxation. Diagonal perturbations, on the other hand, rotate the vector in a horizontal plane and cause dephasing of the wave functions; they cause loss of phase coherence or transverse relaxation. We see that the 5

In fact, the system will relax towards a Boltzmann equilibrium distribution due to a balance with spontaneous emission, which is not included in the present description.

5.2 Quantum dynamics in a non-stationary potential

125

eﬀect of the non-adiabatic coupling vector (previous section) is oﬀ-diagonal: it causes transitions between the two states. In a macroscopic sense we are often interested in the rate of the transition process from state 1 to state 2 (or vice versa). For example, if the two states represent a reactant state R and a product state P (say a proton in the left and right well, respectively, of a double-well potential), the macroscopic transfer rate from R to P is given by the rate constant k in the “reaction” k

R P,

(5.56)

dcR = −kcR + k cP dt

(5.57)

k

fulﬁlling the rate equation

on a coarse-grained time scale. In terms of simulation results, the rate constant k can be found by observing the ensemble-averaged change in population Δρ11 of the R-state, starting at ρ11 (0) = 1, over a time Δt that is large with respect to detailed ﬂuctuations but small with respect to the inverse of k: Δρ11 . (5.58) k=− Δt In terms of the variable z, starting with z = 1, the rate constant is expressed as Δz . (5.59) k=− 2Δt For the two-level system there are analytical solutions to the response to stochastic perturbations in certain simpliﬁed cases. Such analytical solutions can give insight into the ongoing processes, but in simulations there is no need to approximate the description of the processes in order to allow for analytical solutions. The full wave function evolution or – preferably – the density matrix evolution (5.43) can be followed on the ﬂy during a dynamical simulation. This applies also to the multilevel system (next section), for which analytical solutions do not exist. We now give an example of an analytical solution. The perturbation ω(t) is a stochastic vector, i.e., its components are ﬂuctuating functions of time. Analytical solutions can only be obtained when the ﬂuctuations of the perturbations decay fast with respect to the change of r. This is the limit considered by Borgis et al. (1989) and by Borgis and Hynes (1991) to arrive at an expression for the proton transfer rate in a double-well potential; it is also the Redﬁeld limit in the treatment of

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Dynamics of mixed quantum/classical systems

relaxation in spin systems (Redﬁeld, 1965). Now consider the case that the oﬀ-diagonal perturbation is real, with 2C(t) ωy = 0.

(5.60)

ωx =

(5.61)

We also deﬁne the diagonal perturbation in terms of its integral over time, which is a ﬂuctuating phase: t def φ(t) = ωz (t ) dt . (5.62) 0

We start at t = 0 with x, y = 0 and z = 1. Since we consider a time interval in which z is nearly constant, we obtain from (5.55) the following equations to ﬁrst order in t by approximating z = 1: x˙ = yωz ,

(5.63)

y˙ = ωx − xωz ,

(5.64)

z˙ = −yωx .

(5.65)

¿From the ﬁrst two equations a solution for y is obtained by substituting g(t) = (x + iy)eiφ , g(0) = 0,

(5.66)

g˙ = iωx eiφ ,

(5.67)

which yields

with solution

g(t) = i

t

ωx (t )eiφ(t ) dt .

(5.68)

0

¿From (5.66) and (5.68) y(t) is recovered as t y(t) = ωx (t ) cos[φ(t ) − φ(t)] dt .

(5.69)

0

Finally, the rate constant is given by 1 1 ˙ = yωx , k = − z 2 2

(5.70)

which, with τ = t − t and extending the integration limit to ∞ because t is much longer than the decay time of the correlation functions, leads to the following expression: 1 ∞ k= ωx (t)ωx (t − τ ) cos[φ(t) − φ(t − τ )] dτ. (5.71) 2 0

5.2 Quantum dynamics in a non-stationary potential

127

After inserting (5.60), this expression is equivalent to the one used by Borgis et al. (1989): k=

2 2

∞

dτ C(t)C(t − τ ) cos

0

t

ωz (t ) dt .

(5.72)

t−τ

This equation teaches us a few basic principles of perturbation theory. First consider what happens when ωz is constant or nearly constant, as is the case when the level splitting is large. Then the cosine term in (5.72) equals cos ωz τ and (5.72) represents the Fourier transform or spectral density (see Chapter 12, Eq. (12.72) on page 326) of the correlation function of the ﬂuctuating coupling term C(t) at the angular frequency ωz , which is the frequency corresponding to the energy diﬀerence of the two levels. An example is the transition rate between ground and excited state resulting from an oscillating external electric ﬁeld when the system has a nonzero oﬀ-diagonal transition dipole moment μ12 (see (2.92) on page 35). This applies to optical absorption and emission, but also to proton transfer in a double-well potential resulting from electric ﬁeld ﬂuctuations due to solvent dynamics. Next consider what happens in the case of level crossing. In that case, at the crossing point, ωz = 0 and the transfer rate is determined by the integral of the correlation function of C(t), i.e., the zero-frequency component of its spectral density. However, during the crossing event the diagonal elements are not identically zero and the transfer rate is determined by the timedependence of both the diagonal and oﬀ-diagonal elements of the Hamiltonian, according to (5.72). A simplifying assumption is that the ﬂuctuation of the oﬀ-diagonal coupling term is not correlated with the ﬂuctuation of the diagonal splitting term. The transfer rate is then determined by the integral of the product of two correlation functions fx (τ ) and fz (τ ): fx (τ ) = C(t)C(t − τ ), fz (τ ) = cos φ(τ ), with φ(τ ) =

(5.73) t

ωz (t ) dt .

(5.74)

t−τ

For stationary stochastic processes φ is a function of τ only. When ωz (t) is a memoryless random process, φ(τ ) is a Wiener process (see page 253), representing a diﬀusion along the φ-axis, starting at φ = 0, with diﬀusion constant D: ∞ D= ωz (0)ωz (t) dt. (5.75) 0

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Dynamics of mixed quantum/classical systems

This diﬀusion process leads to a distribution function after a time τ of 1 φ2 p(φ, τ ) = √ , (5.76) exp − 4Dτ 4πDτ which implies an exponentially decaying average cosine function: ∞ p(φ, τ ) cos φ dφ = e−Dτ . cos φ(τ ) =

(5.77)

−∞

When the random process ωz (t) is not memoryless, the decay of cos φ will deviate from exponential behavior for short times, but will develop into an exponential tail. Its correlation time D−1 is given by the inverse of (5.75); it is inversely proportional to the correlation time of ωz . The faster ωz ﬂuctuates, the slower cos φ will decay and the smaller its inﬂuence on the reaction rate will be. We end by noting, again, that in simulations the approximations required for analytical solutions need not be made; the reaction rates can be computed by solving the complete density matrix evolution based on time-dependent perturbations obtained from simulations.

5.2.5 The multi-level system The two-state case is able to treat the dynamics of tunneling processes involving two nearby states, but is unable to include transitions to low-lying excited states. The latter are required for a full non-adiabatic treatment of a transfer process. The extension to the multi-level case is straightforward, but the analogy with a three-dimensional rotating top is then lost. The basis functions should be chosen orthogonal, but they need not be solutions of any stationary Schr¨ odinger equation. Nevertheless, it is usually convenient and eﬃcient to consider a stationary average potential and construct a set of basis functions as solutions of the Schr¨ odinger equation with that potential. In this way one can be sure that the basis set adequately covers the required space and includes the ﬂexibility to include low-lying excited states. Mavri and Berendsen (1995) found that ﬁve basis functions, constructed by diagonalization of ﬁve Gaussians, were quite adequate to describe proton transfer over a hydrogen bond in aqueous solution. They also conclude that the use of only two Gaussians is inadequate: it underestimates the transfer rate by a factor of 30! A two-level system can only describe ground-state tunneling and does not allow paths involving excited states; it also easily underestimates the coupling term because the barrier region is inadequately described.

5.3 Embedding in a classical environment

129

In the multi-level case the transfer rate cannot simply be identiﬁed with the course-grained decay rate of the population, such as ρ11 in the two-level case (5.58). This is because a “reactant state” or “product state” cannot be identiﬁed with a particular quantum level. The best solution is to deﬁne a spatial selection function S(r) with the property that it is equal to one in the spatial domain one wishes to identify with a particular state and zero elsewhere. The probability pS to be in that state is then given by (5.78) pS = Ψ∗ ΨS(r) dr = tr (ρS), with

Snm =

φ∗n φm S(r) dr.

(5.79)

5.3 Embedding in a classical environment Thus far we have considered how a quantum (sub)system develops when it is subjected to time-dependent perturbing inﬂuences from classical degrees of freedom (or from external ﬁelds). The system invariably develops into a mixed quantum state with a wave function consisting of a superposition of eigenfunctions, even if it started from a pure quantum state. We did not ask the question whether a single quantum system indeed develops into a mixed state, or ends up in one or another pure state with a certain probability governed by the transition rates that we could calculate. In fact that question is academic and unanswerable: we can only observe an ensemble containing all the states that the system can develop into, and we cannot observe the fate of a single system. If a single system is observed, the measurement can only reveal the probability that a given ﬁnal state has occurred. The common notion among spectroscopists that a quantum system, which absorbs a radiation quantum, suddenly jumps to the excited state, is equally right or wrong as the notion that such a quantum system gradually mixes the excited state into its ground state wave function in the process of absorbing a radiation quantum. Again an academic question: we don’t need to know, as the outcome of an experiment over an ensemble is the same for both views. We also have not considered the related question how the quantum system reacts back onto the classical degrees of freedom. In cases where the coupling between quantum and classical degrees if freedom is weak (as, e.g., in nuclear spins embedded in classical molecular systems), the back reaction has a negligible eﬀect on the dynamics of the classical system and can be disregarded. The classical system has its autonomous dynamics. This is also true for a reaction (such as a proton transfer) in the very ﬁrst beginning,

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Dynamics of mixed quantum/classical systems

when the wave function has hardly changed. However, when the coupling is not weak, the back reaction is important and essential to fulﬁll the conservation laws for energy and momentum. Now it `ıs important whether the single quantum system develops into a mixed state or a pure state, with very diﬀerent strengths of the back reaction. For example, as already discussed on page 112, after a “crossing event” the system “chooses” one branch or another, but not both, and it reacts back onto the classical degrees of freedom from one branch only. Taking the back reaction from the mixed quantum wave function – which is called the mean ﬁeld back reaction – is obviously incorrect. Another example (Tully, 1990) is a particle colliding with a solid surface, after which it either reﬂects back or gets adsorbed. One observes 20% reﬂection and 80% absorption, for example, but not 100% of something in between that would correspond to the mean ﬁeld reaction. It now seems that an academic question that cannot be answered and has no bearing on observations, suddenly becomes essential in simulations. Indeed, that is the case, and the reason is that we have been so stupid as to separate a system into quantum and classical degrees of freedom. If a system is treated completely by quantum mechanics, no problems of this kind arise. For example, if the motion along the classical degree of freedom in a level-crossing event is handled by a Gaussian wave packet, the wave packet splits up at the intersection and moves with certain probability and phase in each of the branches (see, for example, Hahn and Stock (2000) for a wave-packet description of the isomerization after excitation in rhodopsin). But the artiﬁcial separation in quantum and classical parts calls for some kind of contraction of the quantum system to a single trajectory and an artiﬁcial handling of the interaction.6 The most popular method to achieve a consistent overall dynamics is the surface hopping method of Tully (1990), described in Section 5.3.3.

5.3.1 Mean-ﬁeld back reaction We consider how the evolution of classical and quantum degrees of freedom can be solved simultaneously in such a way that total energy and momentum are conserved. Consider a system that can be split up in classical coordinates (degrees of freedom) R and quantum degrees of freedom r, each with its conjugated momenta. The total Hamiltonian of the system is a function of all coordinates and momenta. Now assume that a proper set of orthonormal basis functions φn (r; R) 6

For a review of various methods to handle the dynamics at level crossing (called conical intersections in more-dimensional cases), see Stock and Thoss (2003).

5.3 Embedding in a classical environment

131

ˆ in r-space and is rephas been deﬁned. The Hamiltonian is an operator H resented by a matrix with elements ˆ R, P )|m, Hnm (q, p) = n|H(r,

(5.80)

where P are the momenta conjugated with R. These matrix elements can be evaluated for any conﬁguration (R, P ) in the classical phase space. Using this Hamiltonian matrix, the density matrix ρ evolves according to (5.29) or (5.37), which reduces to (5.35) for basis functions that are independent of the classical coordinates. The classical system is now propagated using the quantum expectation of the forces F and velocities: ˆ F = P˙ = − tr (ρF), with Fnm = n| − ∇R H|m, ˙ = tr (ρ∇P H). R

(5.81) (5.82)

The latter equation is – for conservative forces – simply equal to ˙ = ∇P K, R

(5.83)

˙ for cartesian particle coordinates, because the classical kinetic or V = R energy K is a separable term in the total Hamiltonian. It can be shown (see proof below) that this combined quantum/classical dynamics conserves the total energy of the system: d dEtot = tr (ρH) = 0. (5.84) dt dt This must be considered a minimum requirement for a proper non-stochastic combined quantum/classical dynamics scheme. Any average energy increase (decrease) in the quantum degrees of freedom is compensated by a decrease (increase) in energy of the classical degrees of freedom.7 Note that the force on the classical particles (5.81) is the expectation of the force matrix, ˆ and not the gradient of the which is the expectation of the gradient of H ˆ expectation of H. The latter would also contain a contribution due to the gradients of the basis functions in case these are a function of R. The force calculated as the expectation of the negative gradient of the Hamiltonian is called the Hellmann–Feynman force; the matrix elements Fnm in (5.81) are the Hellmann–Feynman force matrix elements. The use of forces averaged over the wave function is in accordance with Ehrenfest’s principle (see (3.15) on page 43). Note that the energy conservation is exact, even when the basis 7

The DME method with average back reaction has been applied to a heavy atom colliding with a quantum harmonic oscillator (Berendsen and Mavri, 1993) and to a quantum harmonic oscillator in a dense argon gas by Mavri and Berendsen (1994). There is perfect energy conservation in these cases. In the latter case thermal equilibration occurs between the harmonic oscillator and the gas.

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Dynamics of mixed quantum/classical systems

set is incomplete and the wave function evolution is therefore not exact. The forces resulting from gradients of the basis functions are called the Pulay forces; as shown above, they should not be included in the forces on the classical particles when the perturbation term due to the non-adiabatic coupling is included in the dynamics of the quantum subsystem. In the next section we return to the Pulay forces in the adiabatic limit. Proof First we split (5.84) into two parts: dEtot ˙ ˙ = tr (ρH) + tr (ρH), dt

(5.85)

and consider the ﬁrst part, using (5.37): ˙ tr (ρH) =

i ˙ · tr ([ρ, d] H) = R ˙ · tr ([ρ, d] H), tr ([ρ, H] H) + R

(5.86)

because tr ([ρ, H] H) = 0, since tr (ρHH) = tr (HρH).8 The second part of (5.85) can be written out as follows: ˙ =R ˙ · tr (ρ∇R H) + P˙ · tr (ρ∇P H). tr (ρH)

(5.87)

The tricky term is ∇R H: ˆ m + φn |H|∇ ˆ m + φn |∇R H|φ ˆ R φm . ∇R Hnm = ∇R φn |H|φ The middle term equals −Fnm , according to (5.81); in (5.87) it produces a ˙ · P˙ , which exactly cancels the last term in (5.87). The ﬁrst term term −R ˆ can be rewritten with the use of the Hermitian property of H: ∗ ˆ ˆ ˆ R φn )∗ φm ∇R φn |H|φm = dr ∇R φn Hφm = dr (H∇ = (Hd)∗mn = [(Hd)† ]nm , while the third term equals (Hd)nm . Realizing that (Hd)† = d† H† = −dH, the ﬁrst and third term together are equal to [H, d]nm . Thus (5.87) reduces to ˙ =R ˙ · tr (ρ [H, d]) = −R ˙ · tr ([ρ, d] H), tr (ρH) which exactly cancels the term (5.86) left over from the ﬁrst part of (5.85). 8

The trace of a matrix product is invariant for cyclic permutation of the matrices in the product.

5.3 Embedding in a classical environment

133

5.3.2 Forces in the adiabatic limit The considerations in the previous section allow an extrapolation to the pure Born–Oppenheimer approximation (or adiabatic limit) and allow an analysis of the proper forces that act on the classical degrees of freedom in the adiabatic limit, which have been the subject of many discussions. The leading principle, again, is the conservation of total energy. In the Born–Oppenheimer approximation it is assumed that for any point in classical phase space (R, P ) the time-independent Schr¨ odinger equation ˆ = Eψ has been solved exactly for the ground state. The classical coorHψ dinates R are only parameters in this solution. The system is assumed to be and remain in the pure ground state, i.e., in terms of a density matrix with the exact solutions of the Schr¨ odinger equation as basis functions, numbered n = 0, 1, . . ., ρ00 = 1 and ρ˙ = 0. No transitions to excited states are allowed. It is this assumption that forms the crucial approximation. It also implies that the system always remains in its exact ground state, i.e., that it follows adiabatic dynamics. The assumption ρ˙ = 0 implies not only that the Hamiltonian is diagonal, including whatever small perturbations there are, but also that the second ˙ term in (5.35): R[ρ, d], is completely neglected. The ground-state energy E0 (R) functions as the potential energy for the Hamiltonian dynamics of the classical degrees of freedom. Hence the forces must be the negative gradients of the ground-state energy in order to conserve the total energy: ˆ 0. F = −∇ dr ψ0∗ Hψ (5.88) Since all three elements in the integral depend on R, this force is not equal to the Hellmann-Feynman force: ˆ 0. F HF = − dr ψ0∗ ∇Hψ (5.89) The diﬀerence is the Pulay force due to the dependence of the wave function on the nuclear coordinates: ˆ 0 − dr ψ0∗ H∇ψ ˆ F Pulay = − dr ∇ψ0∗ Hψ (5.90) 0 = [d, H]00 , where the last equality follows from the same reasoning as was followed in the proof of (5.84) on page 132. It is not surprising that the Pulay force reappears, because it only canceled in (5.84) against the now neglected term in the density matrix evolution. The Pulay force seems to be a nasty complication, but it isn’t. When

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Dynamics of mixed quantum/classical systems

the Hamiltonian is really diagonal, the term [d, H]00 is equal to zero and the Pulay force vanishes. So, for a pure adiabatic process, the HellmannFeynman forces suﬃce.

5.3.3 Surface hopping dynamics The method of surface hopping (SH), originating from ideas of Pechukas (1969a, 1969b), was introduced by Tully and Preston (1971) and speciﬁed with the minimum number of switches by Tully (1990). The method was designed to incorporate the inﬂuence of excited electronic states into the atomic motion. The basic notion is that there is no single best atomic trajectory subject to the inﬂuence of electronic transitions (which would lead to mean-ﬁeld behavior), but that trajectories must split into branches. The splitting is accomplished by making sudden switches between electronic states, based on the diagonal elements of the quantum density matrix. The density matrix is evolved as usual by (5.35). The sudden switches are determined in each time step Δt by a random process based on the transition probabilities from and to the state before the time step. More precisely, if the present state is n, then consider the rate of change of the population ρnn from (5.35):

i ∗ ∗ ˙ (ρnm Hnm ) − R · (ρnm dnm − ρnm dnm ) (5.91) ρ˙ nn = m =n

2 ∗ ˙ (ρnm Hnm ) − 2(R · ρnm dnm ) , (5.92) = m =n

which can be written as ρ˙ nn =

bnm .

(5.93)

m =n

Now deﬁne a switching probability gnm within a time step Δt from the current state n to other states m as Δtbmn gnm = . (5.94) ρnn If gnm < 0, it is set to zero (i.e., only switches to states with higher probabilities are allowed; this in fact corresponds to the condition of a minimal number of switches). The cumulative probabilities hm = m k=1 gnk are determined. A uniform random number between 0 ≤ ζ < 1 is generated and a switch to state m occurs when hm < ζ < hm+1 . In the vast majority of cases no switch will occur and the system remains

5.3 Embedding in a classical environment

135

in the same state n. The classical forces are now calculated as HellmannFeynman forces from the nth state, not from the complete density matrix. If a switch occurs, one must take ad hoc measures to conserve the total energy: scale the velocity of the classical degrees of freedom (in the direction of the nonadiabatic coupling vector). If the kinetic energy of the particle does not suﬃce to make up for the increase in energy level after the switch, the switch is not made. There are many applications of surface hopping. Mavri (2000) has made an interesting comparison between mean-ﬁeld DME, SH and exact quantum calculation of a particle colliding with a quantum oscillator and found in general that SH is closer to exact behavior than mean-ﬁeld DME, but also concluded that both have their advantages and disadvantages, depending on the case under study. There is nothing in the Schr¨ odinger equation that compels a treatment one way or another. 5.3.4 Other methods The situation regarding the consistent treatment of the back reaction from quantum-dynamical subsystems is not satisfactory. Whereas mean-ﬁeld DME fails to select a trajectory based on quantum probabilities, SH contains too many unsatisfactory ad hoc assumptions and fails to keep track of the coherence between various quantum states. Other approaches have similar shortcomings.9 A possible, but not practical, solution is to describe the system as an ensemble of surface-hopping classical trajectory segments, keeping simultaneously track of the trajectories belonging to each of the states that mix into the original state by DME (Ben-Nun and Martinez, 1998; Kapral and Ciccotti, 1999; Nielsen et al., 2000). The eﬀect of quantum decoherence was addressed by Prezhdo and Rossky (1997a, 1997b). A promising approach is the use of Bohmian particle dynamics (quantum hydrodynamics, see section 3.4 on page 64). The quantum particle is sampled from the initial distribution ψ 2 (r, 0) and moves as a classical particle in an eﬀective potential that includes the quantum potential Q (see (3.93) on page 68). The latter is determined by the wave function, which can be computed either by DME in the usual way or by evaluating the gradient of the density of trajectories. Thus the quantum particle follows a single well-deﬁned trajectory, diﬀerent for each initial sample; the branching occurs automatically as a distribution of trajectories. The back reaction now is 9

There is a vast literature on non-adiabatic semiclassical dynamics, which will not be reviewed here. The reader may wish to consult Meyer and Miller(1979), Webster et al. (1991), Laria et al. (1992), Bala et al. (1994), Billing (1994), Hammes-Schiﬀer (1996), Sun and Miller (1997), M¨ uller and Stock (1998).

136

Dynamics of mixed quantum/classical systems

dependent on the Bohmian particle position. The method has been applied by Lepreore and Wyatt (1999),10 Gindensperger et al. (2000, 2002, 2004) and Prezhdo and Brooksby (2001). The latter two sets of authors do not seem to agree on whether the quantum potential should be included in the classical back reaction. There are still problems with energy conservation11 (which is only expected to be obeyed when averaged over a complete ensemble), and it is not quite clear how the Bohmian particle approach should be implemented when the quantum subsystem concerns some generalized coordinates rather than particles. Although at present the method cannot be considered quite mature, it is likely to be the best overall solution for the trajectory-based simulation of mixed quantum/classical dynamics. Mixed quantum-classical problems may often be approximated in a practical way, when the details of the crossing event are irrelevant for the questions asked. Consider, for example, a chromophore in a protein that uses light to change from one conformation to another. Such systems are common in vision (rhodopsin), in energy transformation (bacteriorhodopsin) and in biological signaling processes. After excitation of the chromophore, the system evolves on the excited-state potential surface (generally going downhill along a dihedral angle from a trans state towards a cis state), until it reaches the conical intersection between excited state and ground state.12 It then crosses over from the excited state to either a trans or a cis ground state, proceeding down-hill. The uphill continuation of the excited state is unlikely, as its probability in the damped, diﬀusion-like multidimensional motion is very small; if it happens it will lead to re-entry into the conical intersection and can be disregarded as an irrelevant process. The fate of the system upon leaving the conical intersection is of much more interest than the details during the crossing event. Groenhof et al. (2004), in a simulation study of the events after light absorption in the bacterial signaling protein PYP (photoactive yellow protein), used a simple approach with a single surface hop from excited to ground state after the excited state was found to cross the ground state. The potential surface of ground and excited states were determined by CASSCF (complete active space SCF, see page 105). After each time step the conﬁguration-interaction vector is determined and it is seen whether the system crossing has occurred. If it has, the classical forces are switched from the excited to the ground state and the system continues 10 11 12

See also Wyatt (2005). See comment by Salcedo (2003). A conical intersection is the multidimensional analog of the two-state crossing, as depicted in Fig. 5.2. The main coordinate in retinal-like chromophores is a dihedral angle or a combination of dihedral angles, but there are other motions, called skeletal deformations, that aid in reaching the intersection.

Exercises

137

on one of the descending branches. Thus the crossing details are disregarded, and hopping between states before or after they cross are neglected. Still, the proper system evolution is obtained with computed quantum yield (fraction of successful evolutions to the cis state) close to the experimentally observed one.

Exercises 5.1

5.2 5.3 5.4

Derive the adiabatic wave functions and energies for two states with diabatic energy diﬀerence ΔE0 and oﬀ-diagonal real coupling energies C (see (5.1)). How do the results diﬀer from those of the previous exercise when the oﬀ-diagonal coupling is purely imaginary? Show that(5.45) and (5.46) imply that the quantity 4ρ12 ρ∗12 + z 2 is a constant of the motion. Prove that (5.77) follows from (5.76).

6 Molecular dynamics

6.1 Introduction In this chapter we consider the motion of nuclei in the classical limit. The laws of classical mechanics apply, and the nuclei move in a conservative potential, determined by the electrons in the Born–Oppenheimer approximation. The electrons are assumed to be in the ground state, and the energy is the ground-state solution of the time-independent Schr¨ odinger equation, with all nuclear positions as parameters. This is similar to the assumptions of Car–Parrinello dynamics (see Section 6.3.1), but the derivation of the potential on the ﬂy by quantum-mechanical methods is far too computeintensive to be useful in general. In order to be able to treat large systems over reasonable time spans, a simple description of the potential energy surface is required to enable the simulation of motion on that surface. This is the ﬁrst task: design a suitable force ﬁeld from which the forces on each atom, as well as the total energy, can be eﬃciently computed, given the positions of each atom.1 Section 6.3 describes the principles behind force ﬁelds, and emphasizes the diﬃculties and insuﬃciencies of simple force ﬁeld descriptions. But before considering force ﬁelds, we must deﬁne the system with its boundary conditions (Section 6.2). The way the interactions over the boundary of the simulated systems are treated is in fact part of the total potential energy description. The force ﬁeld descriptions take the covalent structure of molecules into account. They are not valid when chemical reactions may take place, changing the covalent structure, or the redox state, or even the protonation state of a molecule. In such cases at least the reactive part of the molecular system should be treated diﬀerently, e.g., by quantum-chemical methods. The 1

The name “atom” is used interchangeably with “nucleus;” as the electronic motion is not separately considered, the diﬀerence is immaterial.

139

140

Molecular dynamics

“ab initio molecular dynamics” method (Section 6.3.1) solves the electronic and nuclear equation simultaneously. Methods that use quantum-chemical approaches for a subsystem, embedded in a larger system described by a “standard” force ﬁeld, are called QM/MM methods, and are the subject of Section 6.3.10. The methods to solve the equations of motion are described in Section 6.4. We focus on methods that retain cartesian coordinates for the atomic positions, as those are by far the easiest to implement, and generally also the fastest to execute. Internal constraints are then implemented by special algorithms. A review of the principles of classical mechanics and a more complete description of the special techniques for rigid-body and constrained dynamics is given in Chapter 15. In Section 6.5 coupling of the simulated system to external temperature and pressure baths is described. This includes extended system methods that extend the system with extra degrees of freedom allowing the control of certain variables within the system. Such controls can also be used to invoke non-equilibrium molecular dynamics, driving the system away from thermal equilibrium in a speciﬁed way, and allowing the direct study of transport properties. Some straightforward applications are discussed in Section 6.7. The computation of macroscopic quantities that depend on the extent of accessible phase space rather than on microscopic averages, viz. entropy and free energy, is left for the next chapter.

6.2 Boundary conditions of the system The ﬁrst item to consider is the overall shape of the system and the boundary conditions that are applied. Long-range interactions, notably of electrostatic origin, are dependent on such conditions. Only isolated molecules in the dilute gas phase are an exception, but these have a very limited interest in practice. In general we are concerned with systems consisting of molecules in the condensed phase, with a size much larger than the system we can aﬀord to simulate. Thus the simulated system interacts over its boundaries with an environment that is (very) diﬀerent from vacuum. The important consideration is that the environment must respond to changes in the system; such a response is not only static, involving an average interaction energy, but is also dynamic, involving time-dependent forces reacting to changes in the system.

6.2 Boundary conditions of the system y

6 e

e

141

e

b b j b @ @ b @u e e y i A ηb A A r b b j - - x x a e e ξa e b b b

Figure 6.1 Periodic box (2D, for simplicity) determined by base vectors a and b. Particles i and j in the unit cell are given with their images; the nearest image to i is not j, but j in the NW neighboring cell.

6.2.1 Periodic boundary conditions The simplest, and most often applied, conditions are periodic boundary conditions (Fig. 6.1). The system is exactly replicated in three dimensions, thus providing a periodic lattice consisting of unit cells. Each unit cell can have an arbitrary triclinic shape, deﬁned by three basis vectors a, b, c, with arbitrary angles α, β, γ (α is the angle between b and c, etc.) between the basis vectors. If there is only one angle diﬀerent from 90◦ , the cell is monoclinic; if all angles are 90◦ , the cell is rectangular, and if in addition all basis vectors have equal length, the cell is cubic. Note that the unit cell of a periodic system is not uniquely deﬁned: the origin can be placed arbitrarily, the unit cell can be arbitrarily rotated, and to each base vector a linear combination of integer multiples of other base vectors may be added. The volume of the unit cell does not change with any of these operations. For example, if c is changed into c + na a + nb b (na , nb = 0, ±1, ±2, . . .), the volume (see below) V = (a × b) · c does not change since a and b are both perpendicular to a × b. One can always choose a unit cell for which the projection of b on a is smaller than 12 a, and make sure that γ ≥ 60◦ . Several other cell shapes have been devised, such as the truncated octahedron and the rhombic dodecahedron, both of which pack in three dimensions and have a smaller volume than a cubic cell for the same minimal distance

142

Molecular dynamics

(a)

(b)

(c)

Figure 6.2 (a) Close-packed 2D hexagons, with one of the many possible unit cells describing a corresponding lattice. Each unit cell contains parts of four hexagons and each hexagon is spread over four cells. (b) The 3D rhombic octahedron with 12 faces. (c) A stereo pair depicting a packed array of rhombic dodecahedra and the triclinic unit cell of the corresponding lattice (parallel view: left picture for left eye). Figures reproduced from Wassenaar (2006) by permission of Tsjerk Wassenaar, Groningen.

to a neighboring cell. However, they can all be deﬁned in a triclinic periodic lattice, and there is no need to invoke such special unit cells (Bekker, 1997). Figure 6.2 shows how a periodic, but more near-spherical shape (as the hexagon in two dimensions or the rhombic octahedron in three dimensions) packs in a monoclinic or triclinic box. Coordinates of particles can always be expressed in a cartesian coordinate system x, y, z, but expression in relative coordinates ξ, η, ζ in the periodic unit cell can be convenient for some purposes, e.g., for Fourier transforms

6.2 Boundary conditions of the system

143

(see page 331). These latter are the contravariant components2 of the position vector in the oblique coordinate system with unit vectors a, b, c. r = xi + yj + zk

(6.1)

r = ξa + ηb + ζc.

(6.2)

Here i, j, k are cartesian unit vectors, and a, b, c are the base vectors of the triclinic unit cell. The origins of both coordinate systems coincide. For transformation purposes the matrix T made up of the cartesian components of the unit cell base vectors is useful:3 ⎛ ⎞ ax bx cx T = ⎝ ay by cy ⎠ . (6.3) az bz cz This matrix can be inverted when the three base vectors are linearly independent, i.e., when the volume of the cell is not zero. It is easily veriﬁed that r = Tρ ρ = T

−1

(6.4) (6.5)

r,

where r and ρ denote the column matrices (x, y, z)T and (ξ, η, ζ)T , respectively. These equations represent the transformations between cartesian and oblique contravariant components of a vector (see also page 331). Another characteristic of the oblique coordinate system is the metric tensor g, which deﬁnes the length of a vector in terms of its contravariant components: gij dξi dξj , (6.6) (dr)2 = (dx)2 + (dy)2 + (dz)2 = i,j

where ξi stands for ξ, η, ζ. The metric tensor is given by g = TT T.

(6.7)

Finally, the volume V of the triclinic cell is given by the determinant of the transformation vector: V = (a × b) · c = det T.

(6.8)

This determinant is also the Jacobian of the transformation, signifying the 2

3

One may also deﬁne covariant components, which are given by the projections of the vector onto the three unit vectors, but we shall not need those in this context. We also do not follow the convention to write contra(co)variant components as super(sub)scripts. In cartesian coordinates there is no diﬀerence between contra- and covariant components. In many texts this transformation matrix is denoted with the symbol h.

144

Molecular dynamics

Table 6.1 Unit cell deﬁnitions and volumes for the cubic box, the rhombic dodecahedron and the truncated octahedron (image distance d) Box type

Box volume

cubic

d3

Box a d 0 0

rhombic dodecahedron

⎛ 0.707d3

truncated octahedron

d ⎝ 0 0 ⎛

0.770d3

d ⎝ 0 0

0 d 0

vectors b c 0 0 d 0 0 d ⎞ d/2 ⎠ √d/2 2d/2

√d/3 2 2d/3 0

⎞ −d/3 √ ⎠ √2d/3 6d/3

Box angles α β γ 90◦

90◦

90◦

60◦

60◦

90◦

70.5◦

70.5◦

70.5◦

modiﬁcation of the volume element dx dy dz: dx dy dz =

∂(x, y, z) dξ dη dζ = det T dξ dη dζ. ∂(ξ, η, ζ)

(6.9)

In practice, it is always easier to express forces and energies in cartesian rather than oblique coordinates. Oblique coordinates are useful for manipulation with images. In many applications, notably proteins in solvent, the optimal unit cell has a minimal volume under the condition that there is a prescribed minimum distance between any atom of the protein and any atom of any neighboring image. This condition assures that the interaction between images (which is an artefact of the periodicity) is small, while the minimal volume minimizes the computational time spent on the less interesting solvent. For approximately spherical molecules the rhombic dodecahedron is the best, and the truncated octahedron the second-best choice (see Table 6.1). The easy, and therefore often used, but suboptimal choice for arbitrary shapes is a properly chosen rectangular box. As Bekker et al. (2004) have shown, it is also possible to automatically construct an optimal molecular-shaped box, that minimizes the volume while the distances between atoms of images remain larger than a speciﬁed value. An example is given in Fig. 6.3. For the particular protein molecule shown, and with the same minimum distance between atoms of images, the volumes of the cube, truncated octahedron, rhombic dodecahedron and molecular-shaped boxes were respectively 817,

6.2 Boundary conditions of the system

145

629, 578 and 119 nm3 ; even more dramatically the numbers of solvent (water) molecules were respectively 26 505, 20 319, 18 610 and 3 474, the latter reducing the computational time for MD simulation from 9h41 for the cubic box to 1h25 for the molecular-shaped box. Using a molecular-shaped box in MD simulations, it is mandatory to prevent overall rotation of the central molecule in the box in order not to destroy the advantages of the box shape when time proceeds. An algorithm to constrain overall rotation and translation due to Amadei et al. (2000) can be easily implemented and it has been shown that rotational constraints do not to inﬂuence the internal dynamics of the macromolecule (Wassenaar and Mark, 2006). Artifacts of periodic boundary conditions Periodic boundary conditions avoid the perturbing inﬂuence of an artiﬁcial boundary like a vacuum or a reﬂecting wall, but add the additional artefact of periodicity. Only if one wishes to study a crystal, periodic boundary conditions are natural, but even then they suppress all motions in neighboring unit cells that are diﬀerent from the motions in the central cell. In fact, periodic boundary conditions suppress all phenomena that – in reciprocal space – concern k-vectors that do not ﬁt multiples of the reciprocal basis vectors in (see Section 12.9 on page 331 for a description of the reciprocal lattice) 2π/a∗ , 2π/b∗ , 2π/c∗ . Periodic boundary conditions imply that potential functions are also periodic. For example, the Coulomb energy between two charges q1 (r 1 ), q2 (r 2 ) contains the interaction with all images, including the self-images of each particle. Algorithms to implement lattice sums for Coulomb interactions are described in Chapter 13. This leads to the requirement of overall electrical neutrality of the unit cell to avoid diverging electrostatic energies, to correction terms for the self-energy and to considerable artifacts if the system should correspond to a non-periodic reality. For example, consider two opposite charges at a distance d in the x-direction in a cubic cell with an edge of length a. The Coulomb force is depicted in Fig. 6.4 as a function of d/a. When d/a is not small, the periodicity artefact is considerable, amounting to a complete cancelation of the force at d = a/2 and even a sign reversal. Artifacts of periodicity are avoided if modiﬁed interaction potentials are used that vanish for distances larger than half the smallest length of the unit cell. In this way only the nearest image interactions occur. Of course, this involves a modiﬁcation of the potential that causes its own artifacts, and needs careful evaluation. Care should also be taken with the handling of cutoﬀs and the long-range parts in the potential function, as described on page 159: sudden cut-oﬀs cause additional noise and erroneous behavior; smooth

146

Molecular dynamics

Figure 6.3 Construction of a molecular-shaped triclinic box. The molecule (top left) is expanded with a shell of size equal to half the minimum distance required between atoms of images (top right). Subsequently these shapes are translated into a close-packed arrangement (middle left). Middle right: the unit cell depicted with one molecule including its shel l of solvent. Bottom left: the unit cell as simulated (solvent not shown); bottom right: reconstructed molecules. Figures reproduced by permission of Tsjerk Wassenaar, University of Groningen (Wassenaar, 2006). See also Bekker et al., 2004.

cutoﬀs strongly modify the interaction. There is no good solution to avoid periodicity artifacts completely. The best strategy is to use consistent forces and potentials by inclusion of complete lattice sums, but combine this with

6.2 Boundary conditions of the system

147

100 75 50 single pair lattice sum

← force 25 0

0

−25

−5 energy →

−50

−10 single pair lattice sum

−75 −100

−15

0.2

0.4

−20 0.6 0.8 1 Distance (fraction of box size)

Figure 6.4 The Coulomb energy (black) and the force (grey) between an isolated positive and a negative unit charge at a distance d (solid curves) is compared with the energy and force between the same pair in a cubic periodic box (dashed curves).

studying the behavior of the system as a function of box size. In favorable cases it may also be possible to ﬁnd analytical (or numerical) corrections to the eﬀects of either periodicity or modiﬁcations of the interaction potentials (see the discussion on electrostatic continuum corrections on page 168).

148

Molecular dynamics

6.2.2 Continuum boundary conditions

Other boundary conditions may be used. They will involve some kind of reﬂecting wall, often taken to be spherical for simplicity. The character of the problem may require other geometries, e.g., a ﬂat wall in the case of molecules adsorbed on a surface. Interactions with the environment outside the “wall” should represent in the simplest case a potential of mean force given the conﬁguration of atomic positions in the explicit system. In fact the system has a reduced number of degrees of freedom: all degrees of freedom outside the boundary are not speciﬁcally considered. The situation is identical to the reduced system description, treated in Chapter 8. The omitted degrees of freedom give rise to a combination of systematic, frictional and stochastic forces. Most boundary treatments take only care of the systematic forces, which are derivatives of the potential of mean force. If done correctly, the thermodynamic accuracy is maintained, but erroneous dynamic boundary eﬀects may persist. For the potential of mean force a simple approximation must be found. Since the most important interaction with the environment is of electrostatic nature, the electric ﬁeld inside the system should be modiﬁed with the inﬂuence exerted by an environment treated as a continuum dielectric, and – if appropriate – conducting, material. This requires solution of the Poisson equation (see Chapter 13) or – if ions are present – the Poisson– Boltzmann equation. While for general geometries numerical solutions using either ﬁnite-diﬀerence or boundary-element methods are required, for a spherical geometry the solutions are much simpler. They can be either described by adding a ﬁeld expressed in spherical harmonics, or by using the method of image charges (see Chapter 13). Long-range contributions other than Coulomb interactions involve the r−6 dispersion interaction. Its potential of mean force can be evaluated from the average composition of the environmental material, assuming a homogeneous distribution outside the boundary. Since the dispersion interaction is always negative, its contribution from outside the boundary is not negligible, despite its fast decay with distance. Atoms close to the boundary will feel modiﬁed interactions and thus deviate in behavior from the atoms that are far removed from the boundary. Thus there are non-negligible boundary eﬀects, and the outer shell of atoms must not be included in the statistical analysis of the system’s behavior.

6.3 Force ﬁeld descriptions

149

6.2.3 Restrained-shell boundary conditions A boundary method that has found applications in hydrated proteins is the incorporation of a shell of restrained molecules, usually taken to be spherical, between the system and the outer boundary with a continuum. One starts with a ﬁnal snapshot from a full, equilibrated simulation, preferably in a larger periodic box. One then deﬁnes a spherical shell in which the atoms are given an additional restraining potential (such as a harmonic potential with respect to the position in the snapshot), with a force constant depending on the position in the shell, continuously changing from zero at the inner border of the shell to a large value at the outer border. Outside the shell a continuum potential may be added, as described above. This boundary-shell method avoids the insertion of a reﬂecting wall, gives smooth transitions at the two boundaries, and is easy to implement. One should realize, however, that molecules that would otherwise diﬀuse are now made rigid, and the response is “frozen in.” One should allow as much motion as possible, e.g., restrain only the centers of mass of solvent molecules like water, leaving rotational freedom that allows a proper dielectric response. This and most other boundary methods do not allow elastic response of the environment and could produce adverse building up of local pressure (positive or negative) that cannot relax due to the rigidity of the boundary condition. Examples of spherical boundary conditions are the SCAAS (surface-constrained all-atom solvent) model of King and Warshel (1989), which imposes harmonic position and orientation restraints in a surface shell and treats the shell by stochastic Brownian dynamics, and the somewhat more complex boundary model of Essex and Jorgensen (1995). In general one may question the eﬃciency of boundary methods that are sophisticated enough to yield reliable results (implying a rather extensive water shell around the solute, especially for large hydrated (macro)molecules), compared to periodic systems with eﬃcient shapes, as discussed above. 6.3 Force ﬁeld descriptions The fact that there are many force ﬁelds in use, often developed along different routes, based on diﬀerent principles, using diﬀerent data, specialized for diﬀerent applications and yielding diﬀerent results, is a warning that the theory behind force ﬁelds is not in a good shape. Ideally a force ﬁeld description should consist of terms that are transferable between diﬀerent molecules, and valid for a wide range of environments and conditions. It is often the non-additivity of constituent terms, and the omission of important contributions, that renders the terms non-transferable. Since most

150

Molecular dynamics

force ﬁelds contain parameters adjusted to empirical observations, an error or omission in one term is compensated by changes in other terms, which are then not accurate when used for other conﬁgurations, environments or conditions than those for which the parameters were adjusted. Ideally, ab initio quantum calculations should provide a proper potential energy surface for molecules and proper descriptions for the interaction between molecules. Density functional theory (DFT) has – for small systems – advanced to the point that it is feasible to compute the energy and forces by DFT at every time step of the molecular motion and thus evolve the system dynamically. The “ab initio molecular dynamics” method of Car and Parrinello (1985), described in Section 6.3.1, employs a clever method to solve the electronic and nuclear equations simultaneously. Other quantum approximations that scale more linearly with the number of particles, such as the divide-and-conquer and the tight-binding approximations (see Section 4.9) are candidates for on-the-ﬂy quantum calculations of energies and force during dynamic evolution. In general, however, for large systems such direct methods are not eﬃcient enough and simpler descriptions of force ﬁelds are required. There are several reasons that quantum calculations on isolated molecules do not suﬃce to produce reliable force ﬁelds, and empirical adjustments are still necessary: • For condensed systems, interaction with the (inﬁnite) environment must be properly accounted for, • The force ﬁeld description must necessarily be simpliﬁed, if possible to additive local terms, and this simpliﬁcation involves approximation of the full quantum potential energy surface. • Even high-quality ab initio calculations are not accurate enough to produce overall accuracies better than kB T , as required to yield accurate thermodynamic properties. Note that kB T = 2.5 kJ/mol for room temperature, while an error of 6 kJ/mol in the free energy diﬀerence of two states corresponds to an error of a factor of 10 in concentrations of components participating in an equilibrium. • Small eﬀects that are not incorporated in the Born–Oppenheimer quantum mechanics, such as nuclear quantum eﬀects, must be accounted for. The choice must be made whether or not such corrections are applied to the result of calculations, before parameter adjustments are made. If they are not applied afterwards, the quantum eﬀects are mimicked by adjustments in the force ﬁeld contributions. In fact, this consideration does not

6.3 Force ﬁeld descriptions

151

only apply to quantum corrections, but to all eﬀects that are not explicitly accounted for in the force ﬁeld description. Still, it is through the study of quantum chemistry that insight is obtained in the additivity of constituent terms and the shape of the potential terms can be determined. Final empirical adjustments of parameters can then optimize the force ﬁeld.

6.3.1 Ab-Initio molecular dynamics Considering nuclei as classical point particles and electrons as providing a force ﬁeld for the nuclear motion in the Born–Oppenheimer approximation, one may try to solve the energies and forces for a given nuclear conﬁguration by quantum-chemical methods. The nuclear motion may then be advanced in time steps with one of the standard molecular dynamics algorithms. For eﬃciency reasons it is mandatory to employ the fact that nuclear conﬁgurations at successive time steps are very similar and therefore the solutions for the electronic equations are similar as well. In a seminal article, Car and Parrinello (1985) described the simultaneous solution of the nuclear equations of motion and the evolution of the wave function in a density-functional description. The electron density n(r) is written in terms of occupied single-particle orthonormal Kohn–Sham orbitals (K–S orbitals, see Section 4.7 for a more detailed description); |ψi (r)|2 , (6.10) n(r) = i

where each ψl (r) is a linear combination of well-chosen basis functions. Car and Parrinello chose as basis functions a set of plane waves exp(ik · r) compatible with the periodic boundary conditions. Thus every K–S orbital is a vector in reciprocal space. A point of the Born–Oppenheimer potential energy surface is given by the minimum with respect to the K–S orbitals of the energy functional (4.59): 2 E=− (6.11) dr ψi∗ (r)∇2 ψi (r) + U [n(r); R]. 2me i

Here the ﬁrst term is the kinetic energy of the electrons and the second term is a density functional, containing both the electron-nuclear and electronelectron interaction. The latter consists of electronic Coulomb interactions, and exchange and correlation contributions. The K–S wave functions ψi

152

Molecular dynamics

(i.e., the plane wave coeﬃcients that describe each wave function) must be varied to minimize E while preserving the orthonormality conditions (6.12) dr ψi∗ (r)ψj (r) = δij . The nuclear coordinates are constant parameters in this procedure. Once the minimum has been obtained, the forces on the nuclei follow from the gradient of E with respect to the nuclear coordinates. With these forces the nuclear dynamics can be advanced to the next time step. The particular innovation introduced by Car and Parrinello lies in the method they use to solve the minimization problem. They consider an extended dynamical system, consisting of the nuclei and the K–S wave functions. The wave functions are given a ﬁctitious mass μ and a Lagrangian (see (15.2)) is constructed: 1 ˙ 2 − E(ψ, R). L= dr |ψi |2 + MI R (6.13) I 2 I

This Lagrangian, together with the constraints (6.12), generate the following equations of motion (see Section 15.8): ∂E μψ¨i (r, t) = − + Λik ψk , (6.14) ∂ψi k

¨ I = −∇ MI R RI E ,

(6.15)

where I numbers the nuclei, MI are the nuclear masses and RI the nuclear coordinates. The Λik are Lagrange multipliers introduced in order to satisfy the constraints (6.12). The equations are integrated with a suitable constraint algorithm (e.g., Shake, see Section 15.8). When the ﬁctitious masses of the wave functions are chosen small enough, the wave function dynamics is much faster than the nuclear dynamics and the two types of motion are virtually uncoupled. Reducing the “velocities” and consequently the kinetic energy or the “temperature” of the wave functions will cause the wave functions to move close to the B–O minimum energy surface (in the limit of zero temperature the exact minimum is reached). In fact, this dynamic cooling, reminiscent of the “simulated annealing” method of Kirkpatrick et al. (1983), is an eﬀective multidimensional minimization method. In practice the temperature of the electronic degrees of freedom can be kept low enough for the system to remain close to the B–O energy surface, even when the temperature of the nuclei is considerably higher. Because both systems are only weakly coupled, heat exchange between them is very

6.3 Force ﬁeld descriptions

153

weak. Both systems can be coupled to separate thermostats (see Section 6.5) to stabilize their individual temperatures. There is a trade-oﬀ between computational eﬃciency and accuracy: when the “wave function mass” is small, wave functions and nuclei are eﬀectively uncoupled and the system can remain accurately on its B–O surface, but the electronic motions become fast and a small time step must be used. For larger masses the motions of the electronic and nuclear degrees of freedom will start to overlap and the B–O surface is not accurately followed, but a larger time step can be taken. In any case the Car–Parrinello method is time-consuming, both because a large number of (electronic) degrees of freedom are added and because the time step must be taken considerably smaller than in ordinary molecular dynamics. For algorithmic details and applications of ab initio molecular dynamics the reader is referred to Marx and Hutter (2000).

6.3.2 Simple molecular force ﬁelds The simplest force ﬁelds, useful for large molecular systems, but not aiming at detailed reproduction of vibrational spectroscopic properties, contain the following elements: • Atoms are the mass points that move in the force ﬁeld. In united-atom approaches some hydrogen atoms are “incorporated” into the atom to which they are bound. In practice this is used for hydrogen atoms bound to aliphatic carbon atoms; the resulting CH2 or CH3 groups are “united atoms,” acting as a single moving mass. • Atoms (or united atoms) are also the source points for the diﬀerent terms in the force ﬁeld description. This means that the various contributions to the forces are expressed as functions of the atomic positions. • There are two types of interactions: bonded interactions between dedicated groups of atoms, and non-bonded interactions between atoms, based on their (changing) distance. These two types are computationally diﬀerent: bonded interactions concern atoms that are read from a ﬁxed list, but atoms involved in non-bonded interactions ﬂuctuate and must be updated regularly. Non-bonded interactions are assumed to be pairwise additive. Bonded interactions are of the following types: (i) A covalent bond between two atoms is described by a harmonic po-

154

Molecular dynamics

tential of the form:4 Vb (r i , r j ) = 12 kb (r − b)2 ,

(6.16)

r = |r i − r j |,

(6.17)

where

and kb and b are parameters which diﬀer for each bond type. The harmonic potential may be replaced by the more realistic Morse potential: Vmorse (r i , r j ) = Dij [1 − exp(−βij (rij − b))]2 .

(6.18)

Other forms contain harmonic plus cubic terms. (ii) A covalent bond angle is described by a harmonic angular potential of the form Va (r i , r j , r k ) = 12 kθ (θ − θ0 )2 , where θ = arccos

r ij · r kj , rij rjk

(6.19)

(6.20)

or by the simpler form Va = 12 k (cos θ − cos θ0 )2 .

(6.21)

(iii) Dihedral angles φ are deﬁned by the positions of four atoms i, j, k, l as the angle between the normals n and m to the two planes i, j, k and j, k, l: n·m φ = arccos (6.22) nm where n = r ij × r kj m = r jk × r lk .

(6.23)

The dihedral potential is given by a periodic function Vd (φ) = kφ (1 + cos(nφ − φ0 )).

(6.24)

This makes all minima equal (e.g., the trans and the two gauche states for a threefold periodic dihedral, as between two sp3 carbon atoms). The actual diﬀerence between the minima is caused by the 4

The GROMOS force ﬁeld uses a quartic potential of the form V = (kb b−2 /8)(r 2 − b2 )2 , which for small deviations is virtually equivalent to (6.16), but computationally much faster because it avoids computation of a square root.

6.3 Force ﬁeld descriptions

155

introduction of an extra interaction between atoms i and l, called the 1-4 interaction. Instead of using a 1-4 interaction, one may also use a set of periodic functions with diﬀerent periodicity, or a set of powers of cosine functions, as in the Ryckaert–Bellemans potential VRB (φ) =

5

Cn cosn φ.

(6.25)

n=0

(iv) In order to keep planar groups (as aromatic rings) planar and prevent molecules from ﬂipping over to their mirror images, improper dihedrals are deﬁned, based on four atoms i, j, k, l and given a harmonic restraining potential: Vimproper (ξ) = 12 kξ (ξ − ξ0 )2 .

(6.26)

Bonded interactions, if they are so stiﬀ that they represent high-frequency vibrations with frequency ν kB T /h, can be replaced by constraints. In practice this can only be done for bond length constraints, and in some cases for bond-angle constraints as well (van Gunsteren and Berendsen, 1977). The implementation of constraints is described in Chapter 15, Section 15.8 on page 417. Non-bonded interactions are pair-additive, and a function of the distance rij = r between the two particles of each pair. Pairs that are already involved in bonded interactions are excluded from the non-bonded interaction; this concerns 1-2 and 1-3 interactions along a covalently-bonded chain. The 1-4 interactions are either excluded, or used in modiﬁed form, depending on the dihedral functions that are used. Non-bonded interactions are usually considered within a given cut-oﬀ radius, unless they are computed as full lattice sums over a periodic lattice (only in the case of periodic boundary conditions). They are of the following types: (i) Lennard–Jones interactions describe the short-range repulsion and the longer-range dispersion interactions as C12 C6 − 6, r12 r which can be alternatively expressed as σ 12 σ 6 . vLJ (r) = 4ε − r r vLJ (r) =

(6.27)

(6.28)

The treatment of the long-range part of the dispersion will be separately considered below in Section 6.3.4. The r−12 repulsion term is

156

Molecular dynamics

of rather arbitrary shape, and can be replaced by the more realistic exponential form vrep = A exp(−Br),

(6.29)

which, combined with the dispersion term, is usually referred to as the Buckingham potential. (ii) Coulomb interactions between charges or partial charges on atoms: VC (r) = fel

qi qj εr r

(6.30)

Here fel = (4πε0 )−1 and εr is a relative dielectric constant, usually taken equal to 1, but in some force ﬁelds taken to be a function of r itself (e.g., equal to r measured in ˚ A) to mimic the eﬀect of dielectric screening. The latter form must be considered as ad hoc without physical justiﬁcation. Special care is needed for the treatment of the long-range Coulomb interaction, which is separately described below in Section 6.3.5. The partial charges are often derived from empirical dipole and quadrupole moments of (small) molecules, or from quantum calculations. A simple Mulliken analysis of atomic charges resulting from the occupation of atomic orbitals does not suﬃce; the best partial charges are potential-derived charges: those that reproduce the electric potential in the environment of the molecule, with the potential determined from a high-level ab initio quantum calculation. Once the potential has been determined on a grid, and suitable weight factors are chosen for the grid points, such charges can be found by a least squares optimization procedure. This method suﬀers from some arbitrariness due to the choice of grid points and their weights. Another, more robust, method is to ﬁt the charges to multipoles derived from accurate quantum calculations.5 With the use of pair-additive Coulomb interactions, the omission of explicit polarization, and/or the incomplete treatment of longrange interactions, the empirically optimized partial charges do not and should not correspond to the ab initio-derived charges. The Coulomb interactions should then include the average polarization and the average eﬀects of the omission of polarizing particles in the environment. Modiﬁcation of partial charges cannot achieve the correct results, however, as it will completely miss the dielectric solva5

See Jensen (2006) for a general, and Sigfridsson and Ryde (1998) for a more speciﬁc discussion. The latter authors advocate the multipole-ﬁtting method, as do Swart et al. (2001), who use density-functional theory to derive the multipole moments, and list partial charges for 31 molecules and all aminoacids.

6.3 Force ﬁeld descriptions

157

tion energy of (partial) charges in electronically polarizable environments. If average polarization enhances dipole moments, as in water, the partial charges are enhanced, while reducing the charges may be appropriate to mimic the interactions in a polarizable environment. These deﬁciencies are further discussed in Section 6.3.6.

6.3.3 More sophisticated force ﬁelds Force ﬁelds that go beyond the simple type described above may include the following extra or replacing features: (i) Polarizability This is the single most important improvement, which is further detailed in Section 6.3.6. (ii) Virtual interaction sites Several force ﬁelds use interaction sites that do not coincide with atomic positions. For example, one may place a partial charge at the position of a “bond” midway between two atoms 1 and 2: r = 12 (r 1 + r 2 ). Such sites are always a (vector) function of n atomic positions: r = r(r 1 , r 2 , . . . r n ),

(6.31)

and move with these positions. Virtual sites have no mass and do not participate directly in the equations of motion; they are reconstructed after every dynamic step. The potential energy V (r, . . .) depends on r 1 . . . r n via its dependence on r and the force F acting on a virtual site is distributed among the atoms on which the site depends (we write the α-component): F1α = −

∂r ∂V ∂r · =F · . ∂r ∂x1α ∂x1α

(6.32)

For the simple case of the halfway-site F 1 = F 2 = 12 F . Other linear combinations are similarly simple; more complex virtual sites as outof-plane constructions are more complicated but follow from the same equation. (iii) Dummy particles These are sites that carry mass and participate in the equations of motion. They replace real atoms and are meant to simplify rigid-body motions. The replaced atoms are reconstructed as virtual sites. For example, the 12 atoms with 12 constraints of a rigid benzene molecule C6 H6 can be replaced by three dummy atoms with three constraints, having the same total mass and moments of inertia. All interaction sites (atoms in this case) can be reconstructed

158

(iv)

(v)

(vi)

(vii)

6

Molecular dynamics

by linear combinations from the dummies.6 Feenstra et al. (1999) have used dummy sites to eliminate fast motions of hydrogen atoms in proteins, enabling an increase of time step from the usual 2 fs to as much as 7 fs. Dummy atoms do not really belong in this list because they are not a part of the force-ﬁeld description. Coupling terms Force ﬁelds that aim at accurate reproduction of vibrational properties include coupling terms between bond, bondangle and dihedral displacements. Flexible constraints Internal vibrations with frequencies higher than kB T /h exhibit essential quantum behavior. At very high frequencies the corresponding degrees of freedom are in the ground state and can be considered static. It seems logical, therefore, to treat such degrees of freedom as constraints. However, it is quite tricky to separate the real quantum degree of freedom from the classical degrees of freedom, as the ﬂuctuating force on anharmonic bonds shifts the harmonic oscillator both in position and in energy. Constraining the quantum vibration amounts to constraining the bond length to the position where the net force vanishes. Such “ﬂexible constraints” were ﬁrst proposed by Zhou et al. (2000) and have been implemented in a polarizable ab initio water model by Hess et al. (2002) in molecular dynamics and by Saint-Martin et al. (2005) in Monte Carlo algorithms. The diﬀerence with the usual holonomic constraints is not very large. Charge distributions Descriptions of the electronic charge distributions in terms of point charges is not quite appropriate if accuracy at short distances between sources is required. In fact, the electron distributions in atoms have a substantial width and nearby distributions will interpenetrate to a certain extent. The modiﬁed (damped) shortrange interactions are better represented by charge distributions than by point charges. Exponential shapes (as from the distribution of Slater-type orbitals) are the most appropriate. Multipoles To increase the accuracy of the representation of charge distributions while avoiding too many additional virtual sites, dipoles – and sometimes quadrupoles – may be added to the description. The disadvantage is that the equations of motion become more complicated, as even for dipoles the force requires the computation of electric ﬁeld gradients, due to charges, dipoles and quadrupoles. Dipoles and quadrupoles are subjected to torques, which requires distribution of

The term “dummy” is not always reserved for sites as described here, but is often used to indicate virtual sites as well.

6.3 Force ﬁeld descriptions

potential 0.9

1.0

159

force 1.1

0.9

1.0

1.1

r/rc

r/rc

(a)

0.9

1.0

1.1

0.9

1.0

1.1

r/rc

r/rc

(b) 0.9

1.0

1.1

0.9

1.0

r/rc

1.1

r/rc

(c) Figure 6.5 Truncation schemes for an r−6 dispersion potential. Thick lines: potential (left) and force (right) for (a) truncated potential, (b) truncated force = shifted potential, (c) shifted force. Dashed lines give the correct potential and force.

forces over particles that deﬁne the multipole axes. Its use is not recommended. (viii) QM/MM methods combine energies and forces derived from quantumchemical calculations for selected parts of the system with force ﬁelds for the remainder (see Section 6.3.10). (ix) Ab initio molecular dynamics applies DFT-derived forces during dynamics evolution: see Section 6.3.1.

6.3.4 Long-range dispersion interactions The non-bonded potential and force calculations are usually based on lists of atom or group pairs that contain only pairs within a given distance. The reason for this is a computational one: if all pairwise interactions are included, the algorithm has an N 2 complexity, which runs out of hand for large systems. This implies the use of a cut-oﬀ distance, beyond which the interaction is either neglected, or treated in a diﬀerent way that is less computationally demanding than N 2 . For the r−6 dispersion potential such cut-oﬀs can be applied without gross errors, but for the r−1 Coulomb potential simple cut-oﬀs give unacceptable errors.

160

Molecular dynamics

When an abrupt force and potential cut-oﬀ is used, the force is no longer the derivative of the potential, and therefore the potential is no longer conservative. As illustrated in Fig. 6.5a, the derivative of a truncated potential contains a delta function at the cut-oﬀ radius. Incorporation of this unphysical delta function into the force leads to intolerable artifacts. The use of a truncated force without delta function (Fig. 6.5b) implies that the eﬀective potential function in the simulation is not the truncated real potential but a shifted potential obtained by integration of the truncated force. One may then expect that equilibrium ensembles generated by dynamic trajectories diﬀer from those generated by Monte Carlo simulations based on the truncated potential. But even truncated forces with shifted potentials generate artifacts. When particles diﬀuse through the limit of the interaction range they encounter a sudden force change leading to extra noise, to heating artifacts and to artifacts in the density distributions. The discontinuity of the force causes errors when higher-order integration algorithms are used that rely on the existence of force derivatives. Such eﬀects can be avoided by shifting the force function to zero at the cut-oﬀ distance (Fig. 6.5c), but this has an even more severe inﬂuence on the eﬀective potential, which now deviates from the exact potential over a wide range. Several kinds of switching functions, switching the force smoothly oﬀ at the cut-oﬀ radius, are used in practical MD algorithms. The user of such programs, which seemingly run error-free even for short cut-oﬀs, should be aware of the possible inadequacies of the eﬀective potentials. Of course, the error due to neglect of the long-range interaction beyond the cut-oﬀ radius rc can be reduced by increasing rc . This goes at the expense of longer pair lists (which scale with rc3 ) and consequently increased computational eﬀort. In addition, rc should not increase beyond half the smallest box size in order to restrict interactions to the nearest image in periodic systems. One should seek an optimum, weighing computational eﬀort against required precision. But what is the error caused by the use of a modiﬁed interaction function? Let us consider a homogeneous ﬂuid with an interatomic dispersion interaction v disp = −C6 r−6 and compute the correction terms to energy and pressure for three commonly used short-range interactions v sr : truncated potential, truncated force, shifted force. The average number density is ρ and the radial distribution function is g(r): given the presence of a particle at the origin, the probability of ﬁnding another particle in a volume element dr equals ρ g(r) dr. Noting that the correction Δv(r) involves the full dispersion interaction minus the employed short-range interaction, the

6.3 Force ﬁeld descriptions

161

correction to the potential energy and therefore to the internal energy u per particle is ∞ 1 Δv(r)4πr2 g(r) dr, (6.33) Δu = ρ 2 0 and the pressure correction is 2π ΔP = − ρ2 3

∞

r3 g(r)

0

dΔv(r) dr, dr

(6.34)

as is easily derived from the virial expression for the pressure (see Section 17.7.2 on page 484) 1 r ij · F ij , (6.35) P V = N kb T + 3 i

j>i

with r ij = r i − r j and F ij is the force exerted by j on i: r ij dv(r) . F ij = − dr r=rij rij

(6.36)

We obtain the following results, assuming that g(r) = 1 for r >= rc and that the number of particles within rc 1: (i) Truncated potential v sr = −C6 r−6 , 2π Δu = − ρC6 rc−3 , 3 ΔP = −2πρ2 C6 rc−3 .

(6.37) (6.38) (6.39)

The pressure correction consists for two-thirds of a contribution from the missing tail and for one third of a contribution due to the discontinuity of the potential at the cut-oﬀ radius. (ii) Truncated force v sr = −C6 r−6 + C6 rc−6 , 4π Δu = − ρC6 rc−3 , 3 4π 2 ΔP = − ρ C6 rc−3 . 3

(6.40)

v sr = −C6 r−6 − 6C6 rrc−7 + 7C6 rc−6 , 7π Δu = − ρC6 rc−3 , 3

(6.43)

(6.41) (6.42)

(iii) Shifted force

(6.44)

162

Molecular dynamics

ΔP = −

7π 2 ρ C6 rc−3 . 3

(6.45)

In order to avoid diﬃculties caused by the discontinuity in the potential, even for MC simulations where the pressure is aﬀected, the potential should never be truncated, neither in MD nor in MC. In the derivation of the corrections we have neglected details of the radial distribution function in the integrals, which is justiﬁed when rc extends beyond the region where g(r) diﬀers from 1. Thus Δu depends only – and linearly – on density, with the consequences that the change in Helmholtz free energy equals Δu, that there is no change in entropy, and that ΔP = ρΔu. The eﬀect of the correction is appreciable, especially aﬀecting vapor pressure and the location of the critical point.7 For example, for the Lennard–Jones liquid argon 8 at 85 K, not far from the boiling point (87.3 K), the energy correction for a shifted force with rc = 1 nm is −0.958 kJ/mol, and the pressure correction is −20.32 kJ mol−1 nm−3 = −337 bar! In Table 6.2 the corrections in the thermodynamic properties at this particular density-temperature state point are compared to the thermodynamic values themselves. The eﬀects are non-negligible, even for this rather long cut-oﬀ radius of about 3 σ. For water with9 C6 = 0.002617 and at room-temperature liquid density the shifted force correction for 1 nm cut-oﬀ is −0.642 kJ/mol for the energy and −357 bar for the pressure. Such corrections are essential. Still, they are usually not applied, and the models are parameterized to ﬁt empirical data using MD with a given cut-oﬀ method. It is clear that the model parameters are then eﬀective parameters that incorporate the eﬀect of restrictions of the simulations; such parameters must be readjusted when other cut-oﬀs or long-range methods are applied. While such eﬀective potentials are convenient (commonly they do not only imply the eﬀects of long-range treatment but also of lack of polarizability and other contributions that are not pair-additive, neglect of speciﬁc interaction terms and neglect of quantum eﬀects), they tend to restrict the generality and transferability of force ﬁelds. At this point it is worthwhile to remark that for dispersion (and other power-law interactions) long-range contributions can be evaluated under pe7

8 9

See for the Lennard–Jones equation-of-state and a discussion on truncation of LJ interactions: Nicolas et al. (1979), Smit (1992), Johnson et al. (1993) and Frenkel and Smit (2002). The critical temperature for the truncated force shifts as much as 5% downward when the cut-oﬀ is decreased from 3.5 to 3σ. σ = 0.34 nm; ε = 119.8 kB = 0.9961 kJ/mol; C6 = 4εσ 6 = 0.006155 kJ mol−1 nm6 . Density at 85 K and 1 bar: 35.243 mol/dm3 (number density 21.224 nm−3 ). Value for the SPC (Berendsen et al., 1981) and SPC/E model (Berendsen et al., 1987), which is the value recommended by Zeiss and Meath (1975), based on experimental data.

6.3 Force ﬁeld descriptions

163

Table 6.2 Corrections to thermodynamic properties of liquid argon at 85 K and a density of 35.243 mol/dm−3 , if “measured” by isobaric/isochoric MD with shifted force, cut-oﬀ at 1 nm. The second column gives the corresponding thermodynamic properties of argon (data from the Handbook of Chemistry and Physics (Lide, 1995)) Correction on MD value

Thermodynamic value

Unit

ΔU = −0.958 ΔH = −1.915 ΔS = 0 ΔP = −337 Δμ = −1.915 pcorr sat = 0.067

U = −4.811 H = −4.808 S = 53.6 P =1 μ = −9.634 psat = 1

kJ/mol kJ/mol J mol−1 K−1 bar kJ/mol bar at 87.3 K

Radial distribution function g(r) 3.5 cut-off

3 Na+ – Na+ Na+ – Cl– Cl– – Cl–

2.5 2 1.5 1 0.5 0.25

0.5

0.75

1

1.25

1.5

1.75

Ion–ion distance r (nm) Figure 6.6 The ion-ion radial distribution functions for an aqueous NaCl solution, simulated with a Coulomb cut-oﬀ radius of 1.6 nm by Auﬃnger and Beveridge (1995).

riodic boundary conditions by Fourier methods similar to the mesh methods that have been worked out for Coulombic interactions (see below).10 10

Essmann et al. (1995) describe the implementation of long-range dispersion forces.

164

Molecular dynamics

6.3.5 Long-range Coulomb interactions The Coulomb interactions are considerably longer-ranged than dispersion interactions, but because of overall charge neutrality they tend to cancel at large distances. Important long-range eﬀects due to polarization of the medium beyond the cut-oﬀ radius persist and must be accounted for. The Coulomb interactions can be cut oﬀ at a given distance, but easily produce severe artifacts when the cut-oﬀ concerns full charges instead of dipoles. It has been remarked by several authors11 that ﬂuids containing ions show an accumulation of like ions and a depletion of oppositely charged ions near the cut-oﬀ radius. The ion–ion radial distribution function shows severe artifacts near the cut-oﬀ radius, as depicted in Fig. 6.6. It is understandable that like ions accumulate at the cut-oﬀ: they repel each other until they reach the cut-oﬀ distance, after which they will try to diﬀuse back in. Such eﬀects are in fact intolerable; they do not occur with smooth forces and with electrostatic interactions computed by lattice sums. The use of a straight particle–particle cut-oﬀ is also detrimental when dipolar interactions represented by two opposite charges are considered: when the cut-oﬀ cuts between the two charges of a dipole, a full charge is in eﬀect created and forces ﬂuctuate wildly with distance (see Fig. 6.7). The eﬀect is minimized when using force functions that taper oﬀ smoothly towards the cut-oﬀ radius (with vanishing force and derivative), but of course such forces deviate appreciably from the true Coulomb form. Some force ﬁelds use cut-oﬀs for charge groups that are neutral as a whole, rather than for individual partial charges. The eﬀect of dielectric response of the medium beyond the cut-oﬀ radius can be incorporated by the introduction of a reaction ﬁeld.12 It is assumed that the medium outside the cut-oﬀ radius rc has a relative dielectric constant εRF and – if applicable – an ionic strength κ. We ﬁrst assume that the system contains no explicit ions and truncation is done on a neutral-group basis; ionic reaction ﬁelds will be considered later. When a spherical force truncation is used, any charge qi will fully interact with all other charges qj within the cut-oﬀ range rc , but misses the interaction with (induced) dipoles and charge densities outside this range. The latter can be added as a potential of mean force, obtained by integrating the forces due to the reaction 11

12

Brooks et al. (1985) consider the eﬀects of truncation by integral equation techniques and by Monte Carlo simulations; Auﬃnger and Beveridge (1995) apply MD with truncation; Tironi et al. (1995) compare truncation with other long-range techniques. See Chapter 13, (13.82) on page 347 and (13.87) on page 347 for the derivation of reaction ﬁelds. For application to simulation see the original paper (Barker and Watts, 1973) and Barker (1994); Tironi et al. include ionic strength, and H¨ unenberger and van Gunsteren (1998) compare diﬀerent reaction ﬁeld schemes.

6.3 Force ﬁeld descriptions

165

Force on charge pair 1

0.1 nm

0.1 nm

− +

r

− +

0.5 0 −0.5 −1 0.7

0.8

0.9 1 1.1 1.2 Center-to-center distance r (nm)

Figure 6.7 The eﬀect of various cut-oﬀ methods on the force acting between two pairs of charges. Thick line: truncation on the basis of 1 nm group cut-oﬀ; thick line with dashed extension: exact force. Thin line: truncation based on 1 nm atomic cut-oﬀ. Dash–dot line: shifted force 1/r2 − 1/rc2 with 1 nm atomic cut-oﬀ. Dotted line: force from cubic spread function, see page 369 and (13.194), with 1 nm atomic cut-oﬀ, meant to be completed with long-range component.

ﬁeld from the medium outside rc . We assume that the interactions within range are based on a group cut-oﬀ, and denote the inclusion condition by Rij ≤ rc ; Rij is the distance between the reporter positions of the neutral groups to which i and j belong. Pairs that are excluded from non-bonded short-range interactions, because their interactions are already accounted for in other bonded force-ﬁeld terms, are indicated to belong to an exclusion list exclst. We need to deﬁne the total dipole moment in the sphere M : qj (r j − r i ). (6.46) M= j;Rij ≤rc

The reaction ﬁeld at the center of the sphere, i.e., at position r i , is determined by M and given by (13.87) on page 347: E RF (r i ) =

1 f (εr , κ) qj (r j − r i ), 3 4πε0 rc

(6.47)

j;Rij ≤rc

2(ε

− 1) , (2ε + 1) κ2 rc2 , ε = εr 1 + 2(1 + κrc )

f (εr , κ) =

(6.48) (6.49)

166

Molecular dynamics

where εr is the relative dielectric constant of the medium outside the cut-oﬀ range, and κ is the inverse Debye length (see (13.55) on page 342). Note that the central charge qi should be included in the sum. The total dipole moment within the sphere gives a ﬁeld in the origin; the total quadrupole moment gives a ﬁeld gradient, etc. When the system contains charges only, and no explicit higher multipoles, we need only the reaction ﬁeld to compute the forces on each particle.13 Including the direct interactions, the force on i is given by ⎡ ⎤ Fi =

qi 4πε0

⎢ ⎢ ⎢ ⎣

j; R ij ≤r c

(i,j)∈exclst /

qj

r ij ⎥ r ij ⎥ − q f (ε , κ) ⎥. j r 3 3 rc ⎦ rij j

(6.50)

Rij ≤rc

def

Here, r ij = r i − r j , hence the minus sign in the reaction ﬁeld term. Note that the inclusion of the reaction ﬁeld simply modiﬁes the force function; in the tin-foil or conducting boundary condition (εr = ∞ or κ = ∞ : f = 1) the modiﬁcation produces a shifted force, which is continuous at the cut-oﬀ radius. This shifted force is well approximated for media with high dielectric constant, such as water with f (εr , κ) = 0.981. Because of the smooth force the RF modiﬁcation yields acceptable dynamics, even in cases where a reaction ﬁeld is not appropriate because of anisotropy and inhomogeneity of the medium. The potential energy function that generates the forces of (6.50) is obtained by integration: ⎤ ⎡ V (r) =

1 ⎢ ⎢ qi ⎢ 4πε0 ⎣ i

j>i; R ij ≤r c

(i,j)∈exclst /

−1 qj (rij − rc−1 ) +

j≥i

qj

⎥ f (εr , κ) 2 2 ⎥ (r − r ) ij c ⎥. 2rc3 ⎦

Rij ≤rc

(6.51) The forces are not the exact derivatives of this potential because the truncation of F ij is based on distances Rij between reporter positions of groups, which depend not only on rij , but also on the position of other particles. Since Rij ≈ rij , this eﬀect is small and is neglected. In addition, there are discontinuities in the reaction-ﬁeld energies when dipoles cross the cut-oﬀ boundary, which lead to impulsive contributions to the forces. Since these are not incorporated into the forces, the eﬀective (i.e., integrated-force) po13

See Tironi et al. (1995) and H¨ unenberg and van Gunsteren (1998) for the full multipole equations. Note that force ﬁelds with explicit dipoles need to consider the reaction ﬁeld gradient as well!

6.3 Force ﬁeld descriptions

167

tentials will slightly deviate from the real reaction-ﬁeld potentials. This situation is similar to the eﬀect of truncated long-range dispersion potentials as discussed above (page 159). The reaction ﬁeld addition does not account for the polarization eﬀects in the medium beyond the cut-oﬀ due to charges rather than dipoles. Naively one may say that the Born reaction potential of a charge (see (13.72) on page 345) is a constant potential with zero gradient, which does not lead to extra forces on the charges and will therefore not inﬂuence the equilibrium distributions. Therefore the Born correction can be applied afterwards to the results of a simulation. However, this reasoning disregards the discontinuities that occur in the potential when two charges enter or leave the interaction range. When the impulsive derivative of such potentials are not incorporated into the force, the eﬀective potential – which is the integral of the forces – does not equal the Born correction. For charges these eﬀects are more severe than for dipoles. Let us consider the simple case of two charges. The Born-corrected, rc -truncated, interaction energy for a set of charges is: ⎡ ⎤ U =

qj 1 ⎢ 1 g(ε, κ) ⎥ ⎥, qi ⎢ − q j ⎣ ⎦ 4πε0 r 2 r ij c j>i j i

rij ≤rc

(6.52)

rij ≤rc

1 def . g(ε, κ) = 1 − εr (1 + κrc )

(6.53)

Applying this to two charges q1 and q2 at a distance r, we ﬁnd for the potential energy: 1 1 2 g(ε, κ) (q1 + q22 ) , r > rc : V (r) = V∞ = − 4πε0 2 rc

q1 q2 1 1 2 g(ε, κ) r ≤ rc : V (r) = − (q1 + q2 ) 4πε0 r 2 rc 1 = V∞ + q1 q2 (r−1 − g(ε, κ)rc−1 ). 4πε0

(6.54)

The eﬀective potential obtained from the integrated truncated force would simply yield a shifted potential: Veﬀ (r) =

1 q1 q2 (r−1 − rc−1 ). 4πε0

(6.55)

It is interesting that in conducting-boundary conditions (g(ε, κ = 1) the potential equals a shifted potential, plus an overall correction V∞ equal to the sum of the Born energies of the isolated charges. While reaction ﬁelds can be included in the force ﬁeld with varying degrees

168

Molecular dynamics

of sophistication,14 they are never satisfactory in inhomogeneous systems and systems with long-range correlation. The latter include in practice all systems with explicit ions. The polarization in the medium is not simply additive, as is assumed when reaction ﬁelds are included per particle. For example, the ﬁeld in the medium between a positive and a negative charge at a distance larger than rc is strong and induces a strong polarization, while the ﬁeld between two positive charges cancels and produces no polarization. The total polarization energy is then (in absolute value) larger, resp. smaller than predicted by the Born-corrected force ﬁeld of (6.52), which is indiﬀerent for the sign of the charge. One is faced with a choice between inaccurate results and excessive computational eﬀort. There are three ways out of this dilemma, neither using a reaction ﬁeld correction. The ﬁrst is the application of continuum corrections, treated below. The second is the use of the fast multipole method (FMM), which relies on a hierarchical breakdown of the charges in clusters and the evaluation of multipole interactions between clusters. The method is implemented in a few software packages, but is rather complex and not as popular as the lattice summation methods. See the discussion in Section 13.9 on page 362. The third, most recommendable method, is the employment of eﬃcient lattice summation methods. These are, of course, applicable to periodic systems, but even non-periodic clusters can be cast into a periodic form. There are several approaches, of which the accurate and eﬃcient smooth-particle mesh-Ewald (SPME) method of Essmann et al. (1995) has gained wide popularity. These methods are discussed at length in Chapter 13, Section 13.10 on page 362, to which the reader is referred. Continuum correction methods are due to Wood (1995).15 Consider a “model world” in which the Hamiltonian is given by the force ﬁeld used, with truncated, shifted or otherwise modiﬁed long-range Coulomb interaction, and possibly periodic boundary conditions. Compare with a “real world” with the full long-range interactions, possibly of inﬁnite extension without periodic boundary conditions. Now assume that the diﬀerence in equilibrium properties between the two worlds can be computed by electrostatic continuum theory, since the diﬀerence concerns long-range eﬀects on a scale much coarser than atomic detail, and at such distances from real 14 15

See for advanced reaction ﬁelds, e.g., Hummer et al. (1996), Bergdorf et al. (2003) and Christen et al. (2005). These methods are based on earlier ideas of Neumann (1983), who applied continuum methods to interpret simulation results on dielectric behavior. Several authors have made use of continuum corrections, and most applications have been reviewed by Bergdorf et al. (2003), who considered the eﬀects of truncation, reaction ﬁeld functions and periodic boundary conditions on ionic hydration and on the interaction between two ions in a dielectric medium.

6.3 Force ﬁeld descriptions

169

charges that the dielectric response can be assumed to be linear and ﬁeldindependent. Now correct the simulation results of the model world with the diﬀerence obtained by continuum theory. Separate corrections can be obtained for charge–dipole and dipole–dipole cutoﬀs, and for the eﬀect of periodic boundary conditions. The principle is as simple as that, but the implementation can be quite cumbersome and not applicable to all possible cases. The principle of the method is as follows. Consider a system of sources, taken for simplicity to be a set of charges qi at positions r i , in a dielectric medium with linear (i.e., ﬁeld-independent) local dielectric constant ε = εr ε0 and without electrostriction (i.e., ﬁelddependent density). The local polarization P (r) (dipole density) is given by a product of the (local) electric susceptibility χ and the electric ﬁeld E:16 P = ε0 χE,

χ = εr − 1,

(6.56)

The ﬁeld is determined by the sum of the direct ﬁeld of the sources and the dipolar ﬁelds of the polarizations elsewhere: G(r − r i ) + T (r − r )P (r ) dr . (6.57) E(r) = i

Here G(r) is the ﬁeld produced at r by a unit charge at the origin, and T (r) is the tensor which – multiplied by the dipole vector – yields the ﬁeld at r due to a dipole at the origin. The vector function G equals minus the gradient of the potential Φ(r) by a unit charge at the origin. For G and T one can ﬁll in the actual truncated or modiﬁed functions as used in the simulation, including periodic images in the case of periodic conditions. For example, for a truncated force, these inﬂuence functions are: 1 r for r ≤ rc , 4πε0 r3 = 0 for r > rc , 1 = (3xα xβ − r2 δαβ )r−5 for r ≤ rc , 4πε0 = 0 for r > rc .

G(r) =

T αβ

(6.58)

(6.59)

The ﬁrst term in (6.57) is the vacuum ﬁeld of the set of charges. The integral equation (6.56) with (6.57) can then be numerically solved for P and E and from there energies can be found. The total electrostatic energy is 12 εE 2 dr, which diverges for point charges and must be corrected for the vacuum self-energy (see (13.42) on page 340); the polarization energy is 16

See Chapter 13 for the basic electrostatic equations.

170

Molecular dynamics

Table 6.3 Continuum corrections to the simulated hydration free energy of a sodium ion in 512 water molecules in a periodic box, simulated with Coulomb forces truncated at rci−w for ion-water interactions and rcw−w for water–water interactions. Column 3 gives the simulation results (Straatsma and Berendsen, 1988), column 4 gives the Born correction resulting from the ion-water cutoﬀ, column 5 gives the Born-corrected results, column 6 gives the continuum correction resulting from water-water cutoﬀ (Wood, 1995) and column 7 gives the corrected results. Column 8 gives the corrections resulting from periodicity 1 rci−w nm

2 rcw−w nm

3 ΔGsim kJ/mol

4 i−w CBorn kJ/mol

5 3+4 kJ/mol

6 Cw−w kJ/mol

7 5+6 kJ/mol

8 CPBC kJ/mol

0.9 1.05 1.2 0.9 1.2

0.9 0.9 0.9 1.2 1.2

−424 −444 −461 −404 −429

−76.2 −65.3 −57.2 −76.2 −57.2

−500 −509 −518 −480 −486

+21.0 +32.5 +42.5 +03.6 +13.9

−479 −477 −476 −477 −472

−0.1 −0.2 −0.2 −0.1 −1.3

given by the interaction of each charge with the induced dipoles: 1 qi P (r) · G(r − r i ) dr. Upol = − 2

(6.60)

i

The factor 12 is implicit in the basic equations (see Chapter 13, Section 13.6 on page 339), but can also be derived from changing the charges from zero to the full charge and integrating the work it costs to increment the charges against the existing polarization ﬁeld. While the solution of the integral equation (which can be transformed into a matrix equation, to be solved by a suitable iterative method, see Bergdorf et al., 2003) is needed for the “model world,” the reference calculation for the “real world” can be cast into a Poisson equation and solved by more elegant methods. Finally, the diﬀerence in solvation energy should be used to correct “model-world” simulations. The continuum approximations are rather poor in the immediate neighborhood of an ion, where speciﬁc solvent structures and dielectric saturation may occur; such regions must be covered correctly by simulation with atomic detail. These regions, however, do not contribute signiﬁcantly to the diﬀerence between “model world” and “real world” because they fall well within the cutoﬀ range of the model potentials. Wood (1995) applied continuum corrections to a series of MD simulations by Straatsma and Berendsen (1988) on the free energy of hydration of an

6.3 Force ﬁeld descriptions

171

ion, with various values of water–water and water–ion cutoﬀ radii. Because these early simulations were limited in size (one ion and 512 water molecules in a cubic periodic box, with cutoﬀs up to 1.2 nm), they serve well for demonstration purposes because the corrections are substantial. Table 6.3 shows the results of the free energy gained by charging a neon atom to become a Na+ ion. Diﬀerent values for the ion–water and the water–water cutoﬀ were used (columns 1 and 2). When the simulation results (column 3) are corrected for the Born energy (column 4), using the ion–water cutoﬀ, the results (column 5) appear to be inconsistent and depend strongly on the water–water cutoﬀ. The reason for this is that the ion polarizes the water with the result that there is a substantial water–water interaction beyond the water–water cutoﬀ radius. When the continuum correction is applied to the water–water cutoﬀ as well (column 6), the result (column 7) is now consistent for all simulation conditions, i.e., within the accuracy of the simulations of about 3 kJ/mol. The correction for the periodic condition (column 8) appears to be rather small and to increase with the cutoﬀ radius. This is consistent with the more detailed calculations of Bergdorf et al. (2003), who observe that the corrections for periodicity are quite small when applied to truncated potentials, but much larger when applied to full lattice sums. This can be interpreted to mean that the use of full lattice sums enhances the periodicity artifacts. On the other hand, it prevents the – generally worse – artifacts due to truncation.

6.3.6 Polarizable force ﬁelds The dominant and most relevant omission in the usual force ﬁelds is the incorporation of electronic polarizability. The electron distribution around a given nuclear conﬁguration depends on the presence of external electric ﬁelds; in ﬁrst order a dipole moment μ is induced proportional to the (homogeneous) electric ﬁeld E: μ = αE,

(6.61)

where α is the polarizability tensor of the electron distribution.17 In nonhomogeneous ﬁelds there will be induced higher multipoles proportional to ﬁeld gradients; these we shall not invoke as it is much easier to consider small fragments with induced dipoles than large fragments with induced multipoles. For small ﬁelds the induced moments are proportional to the 17

Note that α is to be expressed in SI units F m2 ; it is convenient to deﬁne a modiﬁed polarizdef

ability α = α/(4πε0 ), which has the dimension volume and is to be expressed in m3 .

172

Molecular dynamics

(a)

x +

(b)

x +

(c)

x +

p

U1pol x +

0

x −

4U1pol

Figure 6.8 The non-additivity of polarization energies. (a) a single charge that polarizes a neutral particle causes an induced dipole and a (negative) polarization energy; (b) The ﬁelds of two equal, but oppositely placed, charges compensate each other; there is no polarization energy; (c) When the ﬁelds add up, the polarization energy due to two charges is four times the polarization energy due to one charge.

ﬁeld; for large ﬁelds a signiﬁcant nonlinear hyperpolarizability may occur, usually in the form of a reduction of the induced moment. Polarizability is a non-additive electrical interaction. While the polarization energy of a neutral particle with polarizability α at a distance r from a non-polarizable charge q equals −α q 2 /(8πε0 r4 ) (here α = α/(4πε0 )), the energy is not additive when multiple sources are present. This is demonstrated in Fig. 6.8 for the simple case of a polarizable atom, situated between two (non-polarizable) point charges, each at a distance r from the atom. When both charges have the same sign, the polarization energy is zero because the ﬁeld cancels at the origin; two charges of opposite sign double the ﬁeld, leading to a polarization energy four times the polarization energy caused by a single charge. Thus polarizability cannot be simply incorporated by an extra r−4 -type pair interaction. When polarizable particles interact, as they physically do by electrical interaction between induced dipoles, the ﬁeld on each particle depends on the dipoles on the other particles. Thus one needs to either solve a matrix equation or iterate to a self-consistent solution. The additional computational eﬀort has thus far retarded the introduction of polarizability into force ﬁelds. One may choose to neglect the interaction between induced dipoles, as has been done by a few authors,18 in which case the extra computational eﬀort 18

Straatsma and McCammon (1990a, 1990b, 1991) introduced a non-iterative polarizable force ﬁeld with neglect of mutual interaction between induced dipoles and applied this to free energy calculations. In the case of a xenon atom dissolved in water the polarization free energy equals −1.24 kJ/mol. In this case the applied model is exact and the polarization is due to the ﬂuctuating electric ﬁeld produced by water molecules. The free energy change between a non-polarizable and a non-mutual polarizable empirical model of water was found to be −0.25 kJ/mol. Linssen (1998, see also Pikkemaat et al., 2002) applied a restricted non-mutual polarizability model to the enzyme haloalkane dehalogenase which contains a chloride ion in a

6.3 Force ﬁeld descriptions

173

is negligible. The eﬀect of neglecting the mutual induced-dipole interaction is an exaggeration of the polarization produced by ions: the ﬁeld of neighboring induced dipoles generally opposes the direct Coulomb ﬁeld. Soto and Mark (2002) showed that the polarization energy of an ion in cyclohexane, using non-mutual polarizability, equals 1.5 times the correct Born energy. While non-mutual polarizability does repair the main deﬁciencies of nonpolarizable force ﬁelds (such as ion solvation in non-polar solvents), its use in accurate force ﬁelds is not recommended. If it is used, empirical eﬀective polarizabilities (smaller than the physical values) should be employed that reproduce correct average ionic solvation energies. One may wonder how important the introduction of polarizability will be. If we put some reasonable numbers in the example of Fig. 6.8, the polarization energy of a CH2 group (α = 1.8 × 10−3 nm3 ) at a distance of 0.3 nm from an elementary charge (say, a chlorine ion) equals −15.4 kJ/mol. Linssen (1998) estimated the polarization energy of an internal chloride ion in the enzyme haloalkane dehalogenase as −112 kJ/mol.19 The polarization interaction of a single carbonyl oxygen atom (α = 0.84×10−3 nm3 ) liganded at 0.3 nm distance to a doubly charged calcium ion (as occurs in several enzymes) is as high as −29 kJ/mol. These are not negligible amounts, and polarization cannot be omitted in cases where full, or even partial, charges occur in non-polar or weakly polar environments. In polar environments, such as water, electronic polarization is less important, as it is dominated by the orientational polarization. The latter is fully accounted for by nonpolarizable force ﬁelds. Still, the electronic dipole moment induced in a water molecule (α = 1.44 × 10−3 nm3 ) at a distance of 0.3 nm from an elementary charge (say, a potassium ion) amounts to 40% of the intrinsic dipole moment of water itself! In non-polarizable water models the increased intrinsic dipole moment takes partially care of the ionic polarizing eﬀect, but is not properly dependent on the charge and distance of the nearby ion. Considering these large eﬀects it is quite amazing that one can get away at all with non-polarizable force ﬁelds. The reason for the success of simple non-polarizable empirical force ﬁelds is that the average eﬀects of polarizability are incorporated into phenomenologically adjusted force ﬁelds. For example, water models that give correct densities, sublimation energies, dielectric constant and structure for the liquid state, all have considerably

19

hydrophobic environment. Soto and Mark (2002) investigated the eﬀect of polarizability on the stability of peptide folding in a non-npolar environment and used a non-mutual polarization model. This value may overestimate the full polarization interaction by a factor of 1.5, as discussed above. Without polarization the ion is far too weakly bound and in simulations it is quickly expelled from its binding site into the external solution.

174

Molecular dynamics

Second virial coefficient (cm 3/mol) 0 -200 -400

experimental MDCHO

-600

MDCHO qu-corr TIP4P

-800

SPC SPC/E (1)

-1000 -1200

SPC/E (2)

300

400

500

600

700

800 900 1000 temperature (K)

Figure 6.9 The second virial coeﬃcients for water for four models (symbols), compared to experimental values (solid curve). TIP4P, SPC and SPC/E are nonpolarizable eﬀective pair potentials; MCDHO is an ab initio polarizable model, given with and without quantum correction. The experimental curve is a cubic spline ﬁtted to the average of seven data sets dated after 1980 (Millot et al., 1998, and those quoted by Saint-Martin et al., 2000). Data for TIP4P and SPC/E (1) are from Kusalik et al. (1995), for MCDHO from Saint-Martin et al. (2000) and for SPC and SPC/E (2) from Guissani and Guillot (1993).

enhanced dipole moments. These enhanced moments are likely to be close to the real average dipole moment (intrinsic plus induced) in the liquid state. The solid state is not very diﬀerent and may also be reasonably represented, but the liquid-state models are accurate neither for the dilute gas phase nor for non-polar environments. Thus non-polarizable force ﬁelds are based on eﬀective pair potentials, which can only be valid for environments not too diﬀerent from the environment in which the model parameters were empirically adjusted. Eﬀective pair potentials do not only incorporate the average induced dipole moments, but also incorporate average quantum eﬀects and average non-additivity of repulsion and dispersion contributions. In the case of water, eﬀective pair potentials exaggerate the interaction between iso lated pairs. Thus the hydrogen-bonding energy between two molecules in the dilute gas phase is too large: −27.6 kJ/mol for SPC and −30.1 kJ/mol for SPC/E, compared to −22.6 kJ/mol derived from dimer

6.3 Force ﬁeld descriptions

175

spectroscopy, and −21.0 kJ/mol from accurate quantum calculations. This is also apparent from the second virial coeﬃcient, which deviates from experimental values in the sense that the dimer attraction is too large (Fig. 6.9).20 As noted by Berendsen et al. (1987), there is an inconsistency in the use of eﬀective pair potentials when these incorporate average induced dipoles: the full electrostatic interaction between two molecules is taken into account, while the polarization energy of induced dipoles equals only half the electrostatic energy. Half of the electrostatic energy is “used” for the formation of the induced dipoles (see below for the correct equations). While polarizable models take this factor of two correctly into account, eﬀective pair potentials do not. The heat of vaporization should be corrected with the self-energy of the induced dipoles. It was found that a water model with such corrections (the extended simple point charge model, SPC/E) gives better values for density, radial distribution function, diﬀusion constant and dielectric constant than the same model (SPC) without corrections. On the other hand, the heat of vaporization, and also the free energy change from liquid to gas, is too large, as the molecule retains its enhanced dipole moment also in the gas phase. As seen in Fig. 6.9, the discrepancy from gas phase dimer interaction is even larger than for the “classical” effective pair models. The boiling point should therefore be higher than the experimental value and free energies of solvation of water into non-polar environments should also be too large.21 There is no remedy to these artifacts other than replacing eﬀective potentials with polarizable ones.

20

21

It is through the pioneering work of A. Rahman and F. H. Stillinger in the early 1970s (Rahman and Stillinger, 1971; Stillinger and Rahman, 1972, 1974) that the importance of eﬀective potentials became clear. Their ﬁrst simulation of liquid water used the Ben-Naim-Stillinger (BNS) model that had been derived on the basis of both gas-phase data (second virial coeﬃcient related to the pure pair potentials) and condensed-phase data (ice). This pair potential appeared too weak for the liquid phase and could be improved by a simple scaling of energy. When a modiﬁed version, the ST2 potential (Stillinger and Rahman, 1974), was devised, the notion of an eﬀective potential was already developed (Berendsen et al., 1987). The SPC dipole moment of 2.274 D is enhanced to 2.351 D in SPC/E (compare the gas phase value of 1.85 D). The heat of vaporization at 300 K (41.4 kJ/mol) increases to 46.6 kJ/mol, which decreases the vapor pressure tenfold and increases the boiling point by several tens of degrees. Amazingly, according to Guissani and Guillot (1993), the SPC/E model follows the liquid-vapor coexistence curve quite accurately with critical parameters (Tc = 640 K, ρc = 0.29 g/cm−3 , Pc = 160 bar) close to those of real water (Tc = 647.13 K, ρc = 0.322 g/cm−3 , Pc = 220.55 bar). It does better than SPC with Tc = 587 K and ρc = 0.27 g/cm−3 (de Pablo et al., 1990).

176

Molecular dynamics

6.3.7 Choices for polarizability The ﬁrst choice to be made is the representation of the induced dipoles in the model. We assume that the ﬁxed sources in the non-polarized model are charges qi only.22 One may: • include induced dipoles (dipolar model ), • represent the induced dipoles by a positive and negative charge, connected with a harmonic spring (shell model ),23 or • modify charges at given positions to include induced dipoles (ﬂuctuating charge model ).24 The dipole, or in the second case one of the charges, can be placed on atoms or on virtual points that are deﬁned in terms of atom positions. For the shell model, the other charge is in principle a massless interaction site. In the third case there must be a suﬃcient number of properly placed charges (four for every polarizability). It is highly recommended to use isotropic polarizabilities on each site; in that case the induced dipole points in the direction of the electric ﬁeld and – if iterated to self-consistency – there is no torque μ × E acting on the dipole or the spring. With anisotropic polarizabilities there are torques, also in the shell model where the spring constant would be an anisotropic tensor, which considerably complicates the equations of motion. In the isotropic dipolar model there is a force only, given by the tensor product of dipole moment and ﬁeld gradient: one needs to evaluate the ﬁeld gradient on each dipolar site. In the isotropic shell model there are only charges and no ﬁeld gradients are needed; however, the number of particles doubles per polarizable site and the number of Coulomb interactions quadruples. The ﬂuctuating charge model can accommodate anisotropic polarizabilities, but requires an even larger number of interactions. The question arises if the choice of isotropic polarizabilities is adequate. After all, many molecules possess an anisotropic polarizability, and this can never be generated by simple addition of isotropic atomic polarizabilities. However, the mutual interaction of induced dipoles saves the day: neighboring induced dipoles enhance an external electric ﬁeld when they are lined up 22

23

24

The inclusion of ﬁxed dipoles is a trivial complication, but one should be careful to obtain forces from the correct derivatives of the dipolar ﬁelds. Forces on dipoles include torques, which are to be transferred to the rigid molecular frame in which the dipoles are deﬁned. The shell model idea originates from solid state physics, where an ion is represented by a positive core and a negative shell which carries the interactions of electronic nature. Such models correctly predict phonon spectra. It was ﬁrst used by Dick and Overhauser (1958) on alkali halide crystals. Although used by Berendsen and van der Velde (1972), the ﬂuctuating charge model was ﬁrst introduced into the literature by Zhu et al. (1991) and by Rick et al. (1994). See also Stern et al. (2001).

6.3 Force ﬁeld descriptions

177

in the ﬁeld direction and depress the ﬁeld when they are situated in lateral directions, thus producing an anisotropic overall polarization. In general it is very well possible to reproduce experimental molecular polarizability tensors based on interacting isotropic atomic polarizabilities.25 Consider as example a homonuclear diatomic molecule, with two points with isotropic polarizability α at the nuclear positions, separated by a distance d, in an external ﬁeld E. The ﬁeld at each point includes the dipolar ﬁeld from the other point. For the induced dipole per point in the parallel and perpendicular directions to the molecular axis we obtain, denoting α = α/(4πε0 ): 2μ αE → μ = μ = α E + , (6.62) 3 4πε0 d 1 − 2α /d3 μ⊥ αE , (6.63) μ⊥ = α E − → μ⊥ = 3 4πε0 d 1 + α /d3 resulting in a molecular anisotropic polarizability: 2α 1 − 2α /d3 2α = 1 + α /d3 = (α + 2α⊥ )/3.

α =

(6.64)

α⊥

(6.65)

αiso

(6.66)

The parallel component diverges for a distance of (2α )1/3 . This is a completely unphysical behavior, leading to far too large anisotropies for diatomics (see Fig. 6.10). The remedy (Thole, 1981) is to introduce a damping at short distances by considering the induced dipole not as a point, but as a distribution. Both an exponential decay and a linear decay yield satisfactory results, with a single polarizability for each atom irrespective of its chemical environment, and with a single screening length, if distances between two atoms are scaled by the inverse sixth power of the product of the two atom polarizabilities (van Duijnen and Swart, 1998). Another empirical approach to predict isotropic polarizabilities of organic molecules, based on additivity, uses the number of electrons, their eﬀective quantum numbers and eﬀective nuclear shielding (Glen, 1994). 25

Applequist et al. (1972) included interactions between point dipoles and lists atomic polarizabilities both for a poor additive model and a more successful interaction model. To amend the unrealistic approach to diverging behavior at short distances, Thole (1981) proposed to insert a damping at small distances, as can be realized by considering the polarizable sites not as points, but as smeared-out distributions. A simple triangular distribution with a width scaled by the polarizabilities appeared to yield good results with a single isotropic polarizability for each atom type (H,C,N,O), irrespective of its chemical nature. Van Duijnen and Swart (1998) extended Thole’s model for a wider set of polarizabilities, including ab initio data for 70 molecules and including sulphur and the halogens in addition to Thole’s atom set. They compared a linear as well as an exponential distribution, with a slight preference for the latter.

178

Molecular dynamics

polarizability relative to iso 2.5 1: H2 2: N2

2

3: O2

α// / αiso

4: Cl2 5: F2

1.5 1

2

3

3

4

5

1 α_| ⁄ αiso

0.5

1.4

1.5

1.6

1.7

1.8

1.9 d / α1/3

Figure 6.10 Inadequacy of the interacting dipole model for polarizability of diatomics: The parallel and perpendicular components, relative to the isotropic value of the polarizability are drawn as a function of the internuclear distance, the latter expressed relative to the 1/3 power of the atomic polarizability that reproduces the isotropic molecular polarizability. The symbols indicate the values of the parallel (up-triangles), isotropic (circles) and perpendicular (down-triangles) polarizabilities for diatomic molecules. Filled symbols represent theoretical values from recent NIST tables; open symbols are from experimental values as quoted by Thole (1981).

It should be noted that shell models with interacting shell distributions will be equally capable of producing correct polarizabilities. Let us discuss the question of which description should be preferred in polarizable force ﬁelds: induced dipoles, harmonic shells or ﬂuctuating charges? Computational considerations were given above, and combining a preference for both simplicity and speed, they favor the shell model. More important are scientiﬁc considerations that favor the shell model as well. We follow the argument of Jordan et al. (1995). Consider two neon atoms: if described by a repulsion/dispersion interaction and dipolar polarizabilities centered on the nuclei, there are no electric eﬀects when the atoms approach each other. However, we know that collision-induced infrared absorption can be observed, and the (computed) electrical quadrupole moment of a neon pair is not zero. The logical explanation is that when the electron clouds attract

6.3 Force ﬁeld descriptions

179

each other at larger distances by dispersion, they are pulled in the direction of the other atom and produce a positive quadrupole moment; when they repel each other at shorter distances, they are pushed away and produce a negative quadrupole moment. The quadrupole moment, computed by a high-level ab initio method, does indeed follow the interatomic force. A shell model with the interatomic interaction acting between the shells reproduces such results. Pursuing this idea, Jordan et al. (1995) succeeded in devising a shell model for the nitrogen molecule which reproduces experimental static and dynamic properties in the solid, liquid and gas phases, including a subtle pressure-induced phase transition between a cubic and a hexagonal solid phase. The model contains three shells, one bound to both nitrogens (a “bond”) and the other two bound to one nitrogen (“lone pairs”). Models of this type could be devised for other molecules, but they tend to become complex and computationally intensive. The number of interaction sites would be quite large: one shell per bond and at least one additional shell for each non-hydrogen atom; thus far, complete force ﬁelds along these lines have not been constructed. A somewhat similar and quite successful shell model for water (the MCDHO model) was published by Saint-Martin et al. (2000).26 The model includes a single shell with an exponential charge distribution, which is connected to the oxygen nucleus by a spring but also interacts with the hydrogen charges in the molecule. Parameters were ﬁtted to ab initio calculations of dimer and oligomers. The fact that this model reproduces experimental properties of the gas, liquid and solid phases quite well holds the promise that general force ﬁelds with transferable terms may be derived from ab initio calculations on small molecules, possibly with small empirical corrections, if electronic polarization is properly included. More simple polarizable models have been quite successful as well. Most work has been done on models for water, with the aim to construct models that yield accurate thermodynamic and dynamic properties for a wide range of phases and conditions. While the development of polarizable models is still proceeding, we shall review only the features from which basic principles can be learned. A comprehensive review of models for simulation of water by Guillot (2002) is available.

26

MCDHO: Mobile Charge Densities in Harmonic Oscillators. The article includes Monte Carlo simulations of the liquid state. See also Hess et al. (2002), who performed molecular dynamics on this model.

180

Molecular dynamics

The successful pair-additive simple point charge models27 have been modiﬁed to include polarization, both with induced dipoles and with shell models. The earliest attempt of this type was a modiﬁcation of the SPC model by Ahlstr¨ om et al. (1989), who added a polarizable point dipole on the oxygen atom while reducing the charges to obtain the gas phase dipole moment, and others of this type followed with ﬁne-tuning of parameters.28 Van Maaren and van der Spoel (2001) investigated the properties of a shell model, with the moving charge attached with a spring to a virtual position on the symmetry axis of the molecule about 0.014 nm from the oxygen. They retained the gas phase structure, dipole and quadrupole moment and polarizability, while optimizing the Lennard–Jones interaction. Several water models have a similar position for the negative charge for the simple reason that threecharge models cannot satisfy the experimental quadrupole moment without displacing the negative charge. Yu et al. (2003) developed a simple model with three atoms plus moving charge bound with a spring to the oxygen, which they named the “charge-on-spring” model. The model was intended for computational eﬃciency; it uses a large moving charge and is in fact a shell-implementation of an induced point dipole model. Polarizable models of this kind, whether they use a moving charge or a point dipole, are moderately successful. They repair the main deﬁciencies of eﬀective pair potentials, but do not show the accurate all-phase behavior that one should wish. For example, the interaction in the critical region – and thereby the 27

28

SPC (Berendsen et al., 1981) uses the three atoms as interaction site with partial charges on oxygen (−0.82 e) and hydrogens (+0.41 e). The geometry is rigid, rOH = 0.1 nm; HOH angle = 109.47◦ . There is a Lennard–Jones interaction on the oxygens: C6 = 2.6169 × 10−3 kJ mol−1 nm6 ; C12 = 2.6332×10−6 kJ mol−1 nm12 . The similar TIP3 model (Jorgensen, 1981) has the rigid experimental geometry: OH distance of 0.09572 nm, HOH angle of 104.52◦ , with hydrogen charge of 0.40 e; C6 = 2.1966 × 10−3 ; C12 = 2.4267 × 10−6 (units as above). This model did not show a second-neighbor peak in the radial distribution function and was modiﬁed to TIP3P (Jorgensen et al., 1983), with hydrogen charge of 0.417 e; C6 = 2.4895 × 10−3 and C12 = 2.4351 × 10−6 . In a subsequent four-site model (TIPS2: Jorgensen, 1982) the negative charge was displaced to a point M on the bisectrix in the direction of the hydrogens at 0.015 nm from the oxygen position, while the Lennard–Jones interaction remained on the oxygen, and the parameters were improved in the TIP4P model (Jorgensen et al., 1983). The hydrogen charge is 0.52 e; C6 = 2.5522 × 10−3 and C12 = 2.5104 × 10−6 . Finally, the SPC/E model (Berendsen et al., 1987), which includes an energy correction for average polarization, is like the SPC model but with oxygen charge -0.4238. Van der Spoel et al. (1998) evaluated these models and optimized them for use with a reaction ﬁeld. They conclude that the SPC/E model is superior in predicting properties of liquid water. These include models by Cieplack et al. (1990) with polarizabilities on O and H and extra repulsion/dispersion terms; Caldwell et al. (1990) with SPC/E modiﬁed by O and H polarizabilities and a three-body repulsion term; Dang (1992) with an improvement on the the latter model; Wallqvist and Berne (1993) with a polarizable and a non-polarizable model with extra terms; Chialvo and Cummings (1996) with an evaluation of displacement of negative charge and a point polarizability on oxygen; Svishchev et al. (1996) with the PPC model which has a displaced (by 0.011 nm) negative charge and polarizability only in the molecular plane caused by displacement of the negative charge (this is a partial shell model with enhanced permanent dipole and reduced polarizability), Dang and Chang (1997) with a revised four-site model with the negative charge and dipolar polarizability displaced by 0.0215 nm from the oxygen;

6.3 Force ﬁeld descriptions

181

critical temperature – is underestimated,29 and the dielectric constant is often too large. Both eﬀects are a result of the relatively (too) large polarization interaction at short distances, which is then compensated by (too weak) attraction at longer distances. A shell model developed by Lamoureux et al. (2003) could be made to ﬁt many water properties including the dielectric constant, but only with reduced polarizability (1.04 instead of 1.44 ˚ A3 ). A study of ion hydration (Sp˚ angberg and Hermansson, 2004) with various water models showed too large solvation enthalpies for polarizable models. Giese and York (2004) ﬁnd an overpolarizability for chains of water molecules using screened Coulomb interactions; they suggest that exchange overlap reduces short-range polarization. The answer seems to lie in a proper damping of the short-range Coulombic ﬁelds and polarization. This can be accomplished by Thole-type smearing of charges and polarization, as was introduced into a water model by Burnham et al. (1999), and also by Paricaud et al. (2005), both authors using polarizabilities on all three atoms. However, shell models with charge distributions for the shells are far more natural and eﬃcient. In a revealing study, ﬁtting electrostatic models to ab initio potentials, Tanizaki et al. (1999) showed that point charge models tend to become counterintuitive (as positive charge on lone-pair positions) unless shielding distributed charges are used. As the MCDHO model of Saint-Martin et al. (2000) shows, a single shell for the molecule suﬃces, neglecting the polarizabilities on hydrogen. This model does not only reproduce liquid behavior, but has excellent second virial coeﬃcients (see Fig. 6.9) and excellent liquid–vapor coexistence and critical behavior (Hern´ andez-Cobos et al., 2005). But also with the MCDHO model the polarization is too strong in the condensed phase, leading to a high dielectric constant,30 and some reﬁnement seems necessary.

6.3.8 Energies and forces for polarizable models Consider a collection of ﬁxed source terms, which may be charges qi only or include higher multipoles as well. The sources may be charge or multipole density distributions. They produce an electric ﬁeld E 0 (r) at a position r, which diverges at a point source itself. In addition there will be induced dipoles μsk or shells with charge qks at sites r sk (these may be atoms or virtual sites), with the shell connected by a harmonic spring with spring constant kks to an atom or virtual site r k . The latter may or may not carry a ﬁxed charge as well. These dipoles or shells (which may be points or distributions) 29 30

See Jedlovszky and Richardi (1999) for a comparison of water models in the critical region. Saint-Martin et al. (2005).

182

Molecular dynamics

produce an extra ﬁeld E ind (r) at a position r. The total energy can best be viewed as the sum of the total electrostatic interaction plus a positive polarization energy Vpol needed to create the induced dipoles μ: Vpol =

(μs )2 k

k

2αk

(6.67)

or to stretch the springs Vpol =

1 k

2

kks |r sk − r k |2 .

(6.68)

The total electrostatic energy consists of the source–source interaction Vqq and the source–dipole plus dipole–dipole interaction Vqμ + Vμμ , or for shells the source–shell plus shell–shell interactions Vqs + Vss . In these sums every pair should be counted only once and, depending on the model used, some neighbor interactions must be excluded (minimally a shell has no Coulomb interaction with the site(s) to which it is bound and dipoles do not interact with charges on the same site). The form of the potential function for a pair interaction depends on the shape of the charge or dipole distribution. The polarizations (dipoles or shell positions) will adjust themselves such that the total potential energy is minimized. For dipoles this means: ∂Vtot = 0, Vtot = Vqq + Vqμ + Vμμ + Vpol , ∂μsk

(6.69)

∂Vtot = 0, Vtot = Vqq + Vqs + Vss + Vpol . ∂r sk

(6.70)

or, for shells:

It is easily shown that this minimization leads to s μsk = αk E tot k (r k ),

(6.71)

or, for shells: r sk = r k +

qks tot s E (r k ). kks k

(6.72)

Here E tot is minus the gradient of Vtot . The shell displacement corresponds to an induced dipole moment equal to q s times the displacement and hence corresponds to a polarizability αk =

(qks )2 . kks

(6.73)

6.3 Force ﬁeld descriptions

183

The polarization energy appears to compensate just half of the electrostatic interaction due to induced charges or shells. This, in fact, completes the equations one needs. The forces on site i are determined as minus the gradient to the position r i of the total potential energy.31 The electrostatic interactions between arbitrary charge distributions are not straightforward. The interaction energy between a point charge q at a distance r from the origin of a spherical charge distribution q s w(r) (with ∞ 2 0 4πr w(r) dr = 1) is given (from Gauss’ law) by integrating the work done to bring the charge from inﬁnity to r: ∞ r qq s qq s s 1 s 2 Vqq = 4πr w(r ) dr = φ (r). (6.74) 4πε0 r r2 0 4πε0 In general the function φs (r) and its derivative (to compute ﬁelds) can be best tabulated, using cubic splines for interpolation (see Chapter 19). For a Slater-type exponential decay of the density:32 w(r) =

1 −2r/λ e , πλ3

(6.75)

the potential function is given by

r −2r/λ e . (6.76) φ(r) = 1 − 1 + λ For two interacting charge distributions the two-center integral needs to be tabulated, although Carrillo-Trip et al. (2003) give an approximate expression that gives good accuracy at all relevant distances (note that shells are never very close to each other): Vss ≈

q1s q2s λ1 λ2 + (λ1 + λ2 )2 [1 − (1 + z)e−2z ), z = r12 . r12 (λ1 + λ2 )3

(6.77)

6.3.9 Towards the ideal force ﬁeld General force ﬁelds for molecular systems are not of suﬃcient quality for universal use. They are often adapted to speciﬁc interactions and the terms are not suﬃciently transferable. The latter is clear from the ever increasing number of atom types deemed necessary when force ﬁelds are reﬁned. A large amount of work has been done on models for water, being important, small, polar, polarizable, hydrogen-bonding, and endowed with a huge 31

32

In the dipolar case the induced dipoles are kept constant in taking the derivatives. This is correct because the partial derivatives to the dipole are zero, provided the dipoles have been completely relaxed before the force is determined. λ is the decay length of the corresponding wave function; 32% of the charge density is within a distance λ from the origin.

184

Molecular dynamics

knowledge body of experimental properties. The excessive number of published water models indicate a state of confusion and a lack of accepted guidelines for the development of force ﬁelds. Still, it seems that an advisable approach to constructing accurate molecular models emerges from the present confusion. Let us attempt to clarify the principles involved and make a series of rational choices. We list a few considerations, not necessarily in order of importance: • Simplicity: as such very simple models as SPC/E are already so successful within their realm of applicability, it seems an overkill to devise very sophisticated models with many multipoles and polarizabilities, and many intermolecular interaction terms. However excellent such models may be, they will not become adopted by the simulation community. • Robustness, meaning validity in varying environments. Correct phase behavior can only be expected if the same model represents all phases well, over a wide range of temperature and pressure. The same model should also be valid in very diﬀerent molecular environments. • Transferability, i.e., the principle of constructing the model, and as much as possible also the parameters, should be applicable to other molecules with similar atoms. • Accuracy, meaning the precision reached in reproducing experimental properties. The nature of such properties can be thermodynamic (e.g., phase boundaries, free energies), static structural (e.g., radial distribution functions), static response (e.g., dielectric constant, viscosity, diﬀusion constant) or dynamic response (e.g., dielectric dispersion, spectroscopic relaxation times). The choice of properties and the required accuracy depend on the purpose for which the force ﬁeld is to be used. The goodfor-everything force ﬁeld will be too complex and computationally intensive to be useful for the simulation of large systems with limited accuracy requirements. Thus there is not one ideal force ﬁeld, but a hierarchy depending on the application. • Ab initio computability, meaning that the model parameters should in principle be obtained from ﬁtting to high-level quantum chemistry calculations. This opens the way to construct reliable terms for unusual molecules for which suﬃcient experimental data are not readily available. If ab initio parametrization does not yield acceptable results and empirical reﬁnement is necessary, the model designer should ﬁrst consider the question whether some interaction type or inherent approximation has been overlooked. Ideally, a very accurate parent force ﬁeld, derivable from quantum calcu-

6.3 Force ﬁeld descriptions

185

lations, should be constructed to form the top of the hierarchy. Then, by constraining terms or properties that do ﬂuctuate in the parent model, to their average values in a given set of conditions, simpler child models can be derived with more limited applicability. This process may be repeated to produce even simpler and more restricted grandchildren. In this way simple and eﬃcient force ﬁelds for a limited range of applications may be derived without the need for extensive reparametrization of each new force ﬁeld. This strategy has been advocated by Saint-Martin et al. (2005) and shown to be successful for the MCDHO model.33 The full model has internal ﬂexibility and is polarizable; if the ﬂexibility is constrained at the average value obtained in a given environment and under given conditions, a simpler “child” model emerges that is valid for a range of similar environments and conditions. The same applies to constraining induced dipoles to yield a “grandchild”: a simple four-site eﬀective pair potential, valid in a more limited range of conditions. In a similar fashion child models with simpler force truncation schemes could be constructed, with corrections obtained from the average long-range contributions within the parent model. The parent model should explicitly express separate aspects: any omission will lead to eﬀective incorporation of the omitted aspect into terms of another physical nature at the expense of robustness and transferability of the model. We list a number of these eﬀects: • Quantum character of nuclear motions These can be included in a thermodynamically correct (but dynamically questionable) way by replacing each nucleus by a path integral in imaginary time, approximated by a string of beads as explained in Chapter 5. “Grandparent” models with complete path integral implementations are considerably more complex. More approximately, but suﬃcient for most applications, these quantum eﬀects may be estimated by quantum corrections to second order in , as detailed in Section 3.5, or they may be incorporated in a reﬁnement of the model that includes Feynman–Hibbs quantum widths of the nuclei. Such second-order corrections are not suﬃcient for oscillators with frequencies much above kB T /h. • Quantum character of high-frequency vibrations There are no good methods to include quantum vibrational states dynamically into a classical system. The best one can do is to make an adiabatic approximation and include the equilibrium distribution over quantum states as a potential of mean force acting on the nuclei. Consider an single oscillating bond (the theory is readily extended to coupled vibrations) between two masses m1 33

See also Hern´ andez-Cobos et al. (2005).

186

Molecular dynamics

and m2 , with the bond length as a general coordinate q. Let q0 be the bond length for which there is no net force (potential force plus centrifugal force); q0 depends on the environment and the velocities. The deviation ξ = q − q0 can be separated as a quantum degree of freedom acting in a quadratic potential with its minimum at ξ = 0. It will have an oscillator (angular) frequency ω = k/μ, where k is the force constant and μ the reduced mass m1 m2 /(m1 + m2 ). We now wish to treat the total system as a reduced dynamic system with q constrained to q0 , and omit the quantum degree of freedom ξ. This we call a ﬂexible constraint because the position q0 ﬂuctuates with external forces and angular velocities. In order to preserve the correctness of thermodynamic quantities, the potential of the reduced (i.e., ﬂexibly constrained) system must be replaced by the potential of mean force V mf with respect to the omitted degree of freedom (this is the adiabatic approximation): 1 V mf = Vclass + ω + kB T ln(1 − e−ω/kB T ). 2

(6.78)

Since in principle ω is a function of the classical coordinates through its dependence on the force constant, the potential of mean force will lead to forces on the classical system and energy exchange with the quantum oscillator. However, this dependence is generally small enough to be negligible. In that case it suﬃces to make posteriori corrections to the total energy. The important point is to impose ﬂexible constraints on the system. As Hess et al. (2002), who give a more rigorous statisticalmechanical treatment, have shown, the iterations necessary to implement ﬂexible constraints do not impose a large computational burden when polarization iterations are needed anyway. It should be noted that fully ﬂexible classical force ﬁelds will add an incorrect kB T per oscillator to the total energy. Appropriate corrections are then mandatory. • Intramolecular structural response to external interactions Both full ﬂexibility and ﬂexible constraints take care of intramolecular structural response to external interactions. Generally the structural deviations are small, but they represent a sizeable energetic eﬀect, as the restoring intramolecular potentials are generally quite stiﬀ. Structural ﬂuctuations also react back on the environment and cause a coupling between mechanical forces and polarization. Not only the structural response to external forces should be taken into account, but also the structural response to external electric ﬁelds. • Intramolecular electronic response to external interactions This, in fact,

6.3 Force ﬁeld descriptions

•

•

•

•

187

is the polarizability response, through which the intramolecular charge distribution responds to electrical (and in the case of shell models also mechanical) forces from the environment. Their incorporation in parent models is mandatory. Long-range dispersion forces With reference to the discussion in Section 6.3.4 on page 159, we may state that accurate parent models should include very long range dispersion interactions, preferably by evaluating the corresponding lattice sum for periodic systems. Long-range electrical interactions From the discussion in Section 6.3.5 on page 164 it is clear that parent models must evaluate long-range Coulomb interactions as lattice sums for periodic systems, or else (e.g., for clusters) use no truncation at all. Non-additivity of repulsion and attraction Repulsion between electronic distributions based on exchange is not strictly pair-additive. It is expected that the nonadditivity is at least partly taken care of by repulsions between moving charge distributions, but this remains to be investigated. A parent model should at least have an estimate of the eﬀect and preferably contain an appropriate three-body term. Also the dispersion interaction is nonadditive, but this so-called Axilrod–Teller eﬀect34 is of order r−9 in the distance and probably negligible. Eﬀects of periodicity Parent models should at least evaluate the eﬀects of periodicity and preferably determine the inﬁnite box-size limit for all evaluated properties. Especially for electrolyte solutions eﬀects of periodicity are a matter of concern.

Summarizing all considerations we arrive at a preference for the structure of a suitable parent model. For simplicity, the model should consist of sites (atoms and possibly virtual sites) with partial charges only, without higher multipoles. In order to represent real charge distributions faithfully, smeared charge distributions should be allowed. Polarizability should be realized by moving charge distributions attached to atoms, with special care to correctly represent the induced charge distributions in close proximity of external charges. The reason is not only simplicity, but it is also less likely that induced dipoles or ﬂuctuating charges will eventually prove to be adequate, as they do not include polarization induced by exchange and dispersion forces (see the discussion on page 178). Intermolecular repulsion and dispersion interactions should be largely centered on the moving shells, possibly reﬁned with short-range atom-based corrections. The model should 34

The three-body dispersion is a result of third-order perturbation and is inversely proportional 2 r 2 r 2 (Axilrod and Teller, 1948). to r12 23 13

188

Molecular dynamics

be parameterized by ﬁtting to high-level ab initio calculations, with possible empirical ﬁne tuning.

6.3.10 QM/MM approaches Force ﬁelds of the type described above are not suitable for systems of molecules that undergo chemical transformations. Conﬁgurations along reaction paths diﬀer so much from the covalently bonded stable states that they cannot be described by simple modiﬁcations of the force ﬁelds intended for stable molecules. For such conﬁgurations quantum calculations are required. However, in complex systems in which reactions take place, as in enzymes and other catalysts, the part that cannot be described by force ﬁelds is generally quite limited and it would be an overkill (if at all possible; however, see the ab initio MD treated in Section 6.3.1) to treat the whole system by quantum-chemical methods. For such systems it is possible to use a hybrid method, combining quantum calculations to obtain energies and forces in a limited fragment, embedded in a larger system for which energies and forces are obtained from the much simpler force ﬁeld descriptions (see Fig. 6.11). These methods are indicated as QM/MM (quantum mechanics/ molecular mechanics) methods. The principle was pioneered by Warshel and Levitt (1976), and has been widely applied since the mid-1980s.35 Most simulation packages allow coupling with one or more quantum chemistry programs, which include semi-empirical, as well as density functional and ab initio treatments of a fragment. QM/MM can be used for optimizations, for example of transition states and reaction paths, but also for dynamic trajectories. The forces and energies can be generated every time step, producing a very costly dynamics trajectory, or the QM calculations can be done at selected instances from approximate trajectories. Note that the quantum part is always solved in the Born–Oppenheimer approximation: QM/MM methods do not produce quantum dynamics. The coupling between the QM and the MM part must be carefully modeled. In general, the total energy consists of three contributions, arising from the QM part, the MM part and the interaction between the two. When the 35

For a survey see Gao and Thompson (1998). Chandra Singh and Kollman (1986) introduced the cap atom, which is usually hydrogen and which is kept free to readjust. The coupling between the QM and MM part is described by Field et al. (1990). A layered approach, allowing layers of increasing quantum accuracy, has been introduced by Svensson et al. (1996). In the “AddRemove” scheme of Swart (2003), the capping atom is ﬁrst added on the line pointing to the MM link atom, included in the QM, and then its interaction is removed from the sum of QM and MM energies. Zhang and Yang (1999) use a pseudobond to a one-free-valence atom with an eﬀective core potential instead of a cap atom.

6.4 Solving the equations of motion

189

MM

XQ QM

cap

XM

Figure 6.11 A quantum mechanical fraction embedded in a molecular mechanics environment. Covalent bonds that cross the QM-MM boundary are replaced by a cap atom in the QM part.

QM part consists of a separate molecule, the intermolecular interactions couple the two systems. When the QM part consists of a molecular fragment, the covalent bond(s) between the fragment and the MM environment must be replaced by some other construct, most often a “cap” atom or pseudo-atom. In principle there is a discrepancy between the levels of treatment of the QM part, which includes induction by the electric ﬁelds of the environments, and MM part in the case of non-polarizable MM force ﬁelds. The reader is referred to Bakowies and Thiel (1996) for a detailed evaluation of the various ways the QM and MM systems can interact.

6.4 Solving the equations of motion Given a conservative force ﬁeld, we know the forces acting on atoms and on virtual interaction sites. The forces on virtual sites are ﬁrst redistributed over the atoms from which the virtual sites are derived, so we end up with (cartesian) forces on mass points. The description may contain constraints of bond lengths and possibly bond angles. While it is possible to write the equations of motion in internal coordinates (see Chapter 15), these tend to become quite complicated and the general recommendation for atomic systems is to stick to cartesian coordinates, even in the presence of constraints. Modern constraint methods are robust and eﬃcient (see Section 15.8 on page 417). However, for completely rigid molecules the use of quaternions may be considered (see page 413). Consider a system of N particles with mass mi , coordinates r i and a deﬁned recipe to compute the total potential energy Epot = V (r) and the

190

Molecular dynamics

forces F i (r) = −∇i V (r), given the set of all coordinates. Let coordinates and velocities v i be known at time t. We assume that there are no constraints. The equations of motion are simply Newton’s equations (which are Hamiltonian as well): r˙ i = v i , v˙ i = F i /mi .

(6.79)

The total energy Etot = K + V will be conserved: ∂V dV (r) d 1 dEtot mi v i · v˙ i + · vi = mi v 2i + = dt dt 2 dt ∂r i i i i ∂V vi · F i + (6.80) = · v i = 0. ∂r i i

Properly solving these equations of motion will produce a microcanonical or N, V, E ensemble. In practice there will be errors that cause deviations from the ideal behavior: the ﬁnite time step will cause integration errors and the total energy will not be exactly conserved; errors in forces (e.g., due to truncation) will produce pseudorandom disturbances that cause energies to drift. Since the temperature is determined by the equipartition theorem saying that K = 32 N kB T , the temperature may drift even when equilibrium has been attained. Therefore there are always modiﬁcations to the pure Newtonian equations of motion needed to generate long stable trajectories. The equations of motion are solved in time steps Δt. Three important considerations inﬂuence the choice of algorithm: (i) Time reversibility, inherent in the Newtonian equations of motion, should be conserved. (ii) The generated trajectories should conserve volume in phase space, and in fact also wedge products (area) dq∧dp in general, i.e., the algorithm should be symplectic (see Chapter 17, page 495). This is important to conserve equilibrium distributions in phase space, because deviation from symplectic behavior will produce time-dependent weight factors in phase space. This importance of the symplectic property has been emphasized in the 1990s (Leimkuhler and Reich, 1994; Leimkuhler and Skeel, 1994) and is now widely recognized. (iii) Since the computational eﬀort is completely dominated by the force calculation, methods that use only one force evaluation per time step are to be preferred. This rules out the well-known Runge–Kutta methods, which moreover are also not symplectic and lead to erroneous behavior on longer time scales (Leimkuhler, 1999).

6.4 Solving the equations of motion

191

In the past practice of MD the Gear algorithm has been much used. The Gear algorithm predicts positions and a number of derivatives based on a Taylor expansion of previous values (how many depends on the order of the algorithm); it then evaluates the accelerations from the forces at the predicted position, and corrects the positions and derivatives on the basis of the deviation between predicted and evaluated accelerations. There are several variants and predictor–corrector algorithms of this kind have been applied up to orders as high as eight. They are quite accurate for small time steps but not very stable for larger time steps. When the forces are not very precise, it does not help to use high orders. They are neither time-reversible nor symplectic and have not survived the competition of the simpler, more robust, reversible and symplectic Verlet or leap-frog algorithms.36 The latter and its variants including multiple time-step versions, are derivable by the Reference System Propagator Algorithms (RESPA) method of Tuckerman et al. (1992) that uses simple operator algebra and is outlined below. The original Verlet algorithm (Verlet, 1967) does not use the velocities, and employs a simple discretization of the second derivative: x ¨(t) ≈

x(t − Δt) − 2x(t) + x(t + Δt) , (Δt)2

(6.81)

leading to the predicted position (x stands for every particle coordinate; f (t) = Fi (x(t))/mi is the corresponding force component, evaluated from the positions at time t and – for convenience – divided by the mass) x(t + Δt) = 2x(t) − x(t − Δt) + f (t)(Δt)2 + O((Δt)4 ).

(6.82)

The velocity is found in retrospect from v(t) =

v(t + Δt) − v(t − Δt) + O((Δt)2 ), 2Δt

(6.83)

but plays no role in the evolution of the trajectory. It can be more accurately estimated from37 v(t) =

v(t + Δt) − v(t − Δt) f (t − Δt) − f (t + Δt) + + O((Δt)3 ). (6.84) 2Δt 12

An equivalent scheme is the leap-frog algorithm, which uses positions at integer time steps and velocities halfway in between time steps (Hockney 36

37

Gear algorithms (Gear, 1971) and their variants have been reviewed and evaluated for use in MD including constraints by van Gunsteren and Berendsen (1977) and in relation to the Verlet algorithm by Berendsen and van Gunsteren (1986). Berendsen and van Gunsteren (1986).

192

Molecular dynamics

and Eastwood, 1988). Starting from v(t − 12 Δt) and x(t) the updates are 1 1 v t + Δt = v t − Δt + f (t)Δt, 2 2 1 (6.85) x(t + Δt) = x(t) + v t + Δt Δt. 2 It can be easily shown that this algorithm is equivalent to Verlet’s and will generate the same trajectory, if the velocity v(t − 12 Δt) is started as [x(t) − x(t − Δt)]/(Δt). The velocity at integer time steps can be recovered as the average of the two velocities at half time steps earlier and later, but only to the O((Δt)2 ) precision of (6.83). In several applications, for example when velocity-dependent forces are applied, it is desirable to know the velocity at the time the position is predicted, rather than a time step later. There are several algorithms, equivalent to Verlet, that deliver equal-time velocities. One is Beeman’s algorithm (Beeman, 1976):

2 1 x(t + Δt) = x(t) + v(t)Δt + f (t) − f (t − Δt) (Δt)2 , 3 6

5 1 1 v(t + Δt) = v(t) + f (t + Δt) + f (t) − f (t − Δt) Δt, (6.86) 3 6 6 but the most popular one is the velocity-Verlet algorithm (Swope et al., 1982): 1 x(t + Δt) = x(t) + v(t)Δt + f (t)(Δt)2 , 2 1 (6.87) v(t + Δt) = v(t) + [f (t) + f (t + Δt)]Δt. 2 This algorithm needs the force at the new time step, but there is only one force evaluation per step. Although it is not immediately obvious, all these algorithms are equivalent (Berendsen and van Gunsteren, 1986). The elegant operator technique considers the exponential Liouville operator to evolve the system in time. We start with (17.152) on page 493: z˙ = iLz,

(6.88)

where z is the vector of generalized coordinates and conjugate momenta. We apply (6.88) to the cartesian Newtonian equations (6.79) and introduce the time-diﬀerential operator D: 0 1 x x˙ x , (6.89) = =D ˆ v v v˙ f 0

6.4 Solving the equations of motion

193

where fˆ is an operator acting on x that produces Fi (x)/mi , (i = 1, . . . 3N ). The x, v vector has a length 6N for N particles. The solution is x x x tD (0). (6.90) (0) = U(t) (t) = e v v v The exponential operator U = exp(tD) is time-reversible: U(−t) = U−1 (t),

(6.91)

and this solution is exact and symplectic.38 Unfortunately we cannot solve (6.90) and we have to solve the evolution over small time steps by approximation. First split the operator D into two simple parts: 0 1 0 0 0 1 + = . (6.92) 0 0 fˆ 0 fˆ 0 These two parts do not commute and we can use the Trotter–Suzuki expansion (see Chapter 14, page 386) et(A+B) ≈ e(t/2)A etB e(t/2)A ,

(6.93)

which can be further subdivided into higher products with higher-order accuracy. Substituting the exponential operators A and B by 0 0 0 1 and Uf (t) = t exp Uv (t) = t exp , (6.94) 0 0 fˆ 0 and using a ﬁrst-order expansion of the exponential: x + vt x , = Uv (t) v v x x Uf (t) , = v + ft v

(6.95) (6.96)

we can, for example, split U(Δt) as follows: U(Δt) ≈ Uf (Δt/2)Uv (Δt)Uf (Δt/2).

(6.97)

Writing this out (exercise 1 on page 209) it is easily seen that the velocityVerlet scheme (6.87) is recovered. Concatenating the force operator for successive steps yields the leap-frog algorithm, (6.85). The method is powerful enough to derive higher-order and multiple time-step algorithms. For 38

U is the transformation matrix of the transformation of (x, v)(0) to (x, v)(t). The Jacobian of the transformation, which is the determinant of U, is equal to 1 because of the general rule det[exp(A)] = exp( tr A); the trace of D = 0.

194

Molecular dynamics

example, a double time-step algorithm with short- and long-range forces is obtained (Tuckerman et al., 1992) by applying the propagator Ul (Δt/2)[Us (δt/2)Uv (δt)Us (δt/2)]n Ul (Δt/2),

(6.98)

where Us and Ul are the propagators for the short- and long-range forces, respectively, and Δt = nδt.

6.4.1 Constraints The incorporation of constraints is fully treated in Section 15.8 of Chapter 15, to which we refer. The most popular method is coordinate resetting, as in the routine shake and its variants settle and rattle. In the Verlet algorithm, the coordinate prediction x(t + Δt) is ﬁrst made as if no constraints exist and subsequently the coordinates are iteratively reset in the direction of x(t) until all constraints are satisﬁed. The most robust and stable method is the projection method lincs. The inﬂuence of constraints on the statistical mechanics of canonical averages is treated in Chapter 17, Section 17.9.3 on page 499.

6.5 Controlling the system In almost all cases it is necessary to make modiﬁcations to the Newtonian equations of motion in order to avoid undesirable eﬀects due to the inexact solution of the equations of motion and the inexact evaluation of forces. In most applications it is desirable to simulate at constant temperature, preferably generating a canonical N V T ensemble, and in many applications simulation at constant pressure (N pT ) is preferred above constant volume. In some applications simulation at constant chemical potential is desirable. Finally, a very important class of applications are non-equilibrium simulations, where the system is externally driven out of equilibrium, usually into a steady-state condition, and its response is measured. Depending on the purpose of the simulation it is more or less important to generate an exact ensemble and to know the nature of the ensemble distribution. When the sole purpose is to equilibrate an initially disturbed system, any robust and smooth method that does not require intervention is acceptable. Generally, when only average equilibrium quantities are required, the exact nature of the generated equilibrium ensemble is less important as long as the system remains close to Hamiltonian evolution. One should be aware that there are system-size eﬀects on averages that will depend on the nature of the ensemble (see Chapter 17, Section 17.4.1 on page 462). When

6.5 Controlling the system

195

one wishes to use the properties of ﬂuctuations, e.g., to determine higher derivatives of thermodynamic quantities, knowledge of the exact nature of the generated distribution function is mandatory. Four classes of methods are available to control the system externally: (i) Stochastic methods, involving the application of stochastic forces together with friction forces. They are particularly useful to control temperature. Such forces mimic the eﬀect of elastic collisions with light particles that form an ideal gas at a given temperature. They produce a canonical ensemble. Other types of stochastic control make use of reassigning certain variables (as velocities to control temperature) to preset distribution functions. Stochastic methods in general enforce the required ensemble distribution, but disturb the dynamics of the system. (ii) Strong-coupling methods apply a constraint to the desired quantity, e.g., for temperature control one may scale the velocities at every time step to set the total kinetic energy exactly at the value prescribed by the desired temperature. This is the Gauss isokinetic thermostat. This method follows an old principle by Gauss stating that external constraints should be applied in such a way that they cause the least disturbance. The system dynamics is non-Hamiltonian. The Gauss thermostat produces a canonical distribution in conﬁguration space, but disturbs the dynamical accuracy. (iii) Weak-coupling methods apply a small perturbation aimed at smoothly reducing the property to be controlled to a preset value by a ﬁrst-order rate equation. Such couplings can be applied to velocity scaling to control the temperature and/or to coordinate and volume scaling to control pressure. As the dynamics of the system is nonHamiltonian, weak-coupling methods do not generate a well-deﬁned ensemble. Depending on the coupling strength they generate an ensemble in between microcanonical and canonical for temperature scaling and in between isochoric and isobaric for coordinate scaling. It seems warranted to use ensemble averages but not ﬂuctuations in order to determine thermodynamic quantities. Weak-coupling methods are well suited to impose non-equilibrium conditions. (iv) Extended system dynamics extends the system with extra degrees of freedom related to the controlled quantity, with both a “coordinate” and a conjugate “momentum.” The dynamics of the extended systems remains fully Hamiltonian, which enables the evaluation of the distribution function, but the dynamics of the molecular system is

196

Molecular dynamics

disturbed. A proper choice of extended variables and Hamiltonian can combine the control of temperature and/or pressure combined with a canonical distribution in conﬁguration space. The inﬂuence of such external modiﬁcations on the equilibrium distribution function can be evaluated by the following considerations. First we assume that the underlying (unperturbed) system is Hamiltonian, not only as a diﬀerential equation, but also in the algorithmic implementation. This is never true, and the eﬀects of deviations become apparent only on longer time scales as a result of accumulation of errors. For a true Hamiltonian system, the evolution of density in phase space f (z) is given by the Liouville equation (see Chapter 17, Section 17.8 on page 492, and (17.158)). It can easily be seen that any distribution f (H) in phase space that depends only on the total energy H = K + V will be stationary: df (H) ∂f (H) = z˙ · ∇H = f (H) ∂t dH 3N

i=1

∂H ∂H ∂H ∂H − ∂pi ∂qi ∂qi ∂pi

= 0.

(6.99)

This means that in principle any initial distribution f (H) (e.g., the canonical distribution) will not change in time, but there is no restoring force inherent in the dynamics that will correct any deviations that may (slowly) develop. If an external inﬂuence drives the system to a given distribution, it provides the necessary restoring force. If the Hamiltonian dynamics is accurate, only a small restoring force is needed.

6.5.1 Stochastic methods The application of stochastic disturbances to control temperature goes back to Schneider and Stoll (1978) and corresponds to Langevin dynamics (see Section 8.6); we shall call this method the Langevin thermostat. The idea is to apply a frictional force and a random force to the momenta: p˙i = Fi − γpi + Ri (t),

(6.100)

where Ri (t) is a zero-average stationary random process without memory: Ri (0)Ri (t) = 2mi γi kB T δ(t).

(6.101)

For convenience we now drop the subscript i; the following must be valid for any particle i and the friction and noise can be chosen diﬀerently for every degree of freedom. The random disturbance is realized in time steps Δt and the change in p is a random number drawn from a normal distribution with

6.5 Controlling the system

variance (Δp)2 given by 2 Δt = R(t ) dt (Δp)2 = 0

Δt

dt

197

0

Δt

dt R(t )R(t )

0

= 2mγkB T (Δt).

(6.102)

In fact, it is not required to draw the change in p from a normal distribution, as long as the distribution has zero mean and ﬁnite variance. We now show that this procedure yields extra terms in ∂f /∂t that force the distribution functions of the momenta to which the noise and friction are applied to a normal distribution. The random force on p causes a diﬀusion of p with a diﬀusion constant D given by the mean-square displacement of p (one dimension) in the time interval Δt: (Δp)2 = 2DΔt.

(6.103)

Consequently the diﬀusion constant is given by D = mγkB T.

(6.104)

Diﬀusion leads to Fick’s equation for the distribution function: ∂f (p, t) ∂ 2 f (p, t) =D ∂t ∂p2

(6.105)

and the friction term leads to an extra ﬂux f p˙ and therefore to ∂(f p) ˙ ∂f ∂f =− = γf + γp . ∂t ∂p ∂p

(6.106)

If these two contributions cancel each other, the distribution function will be stationary. The solution of the equation ∂2f ∂f =0 + γf + γp 2 ∂p ∂p

(6.107)

p2 . f (p) ∝ exp − 2mkB T

(6.108)

D is

The total distribution function must be proportional to this normal distribution of pi , and that applies to all momenta to which random and friction forces are applied. Since H contains p2 /2m as an additive contribution, and the Hamiltonian terms will force the stationary distribution function to be a function of H, it follows that the total distribution will be:

H(z) . (6.109) f (z) ∝ exp − kB T

198

Molecular dynamics

Thus the distribution will be canonical and the temperature of the ensemble is set by the relation between applied friction and noise. In a sense the friction and noise driving the canonical distribution compete with accumulating disturbing errors in the numerical solution of the Hamiltonian equations. The applied damping enforces an extra ﬁrst-order decay on velocity correlation functions and thereby disturb the dynamic behavior on time scales comparable to 1/γ. To minimize dynamic disturbance, the γ’s should be taken as small as possible, and need not be applied to all degrees of freedom. But with large errors a strong damping on many particles is required. The Langevin thermostat provides a smooth decay to the desired temperature with ﬁrst-order kinetics. Note that we have given the equations for cartesian coordinates, for which the mass tensor is diagonal and constant; for generalized coordinates the equations become complicated and diﬃcult to handle because the mass tensor is a function of the conﬁguration. The velocity rescaling thermostat of Andersen (1980) has a similar effect, but does not distribute the external disturbance as smoothly as the Langevin thermostat. Andersen’s method consist of reassigning the velocity of a randomly selected molecule at certain time intervals from the appropriate Maxwellian distribution. As expected, a canonical ensemble is generated.

6.5.2 Strong-coupling methods The strong-coupling methods artiﬁcially constrain a property to the desired value, e.g., the total kinetic energy to a prescribed value determined by the desired temperature. This is accomplished by scaling the velocities with a multiplicative factor that preserves the shape of the distribution function. This amounts to adding an acceleration v˙ = −αv t every degree of freedom. This isokinetic or Gauss thermostat was introduced by Hoover et al. (1982) and Evans (1983). Following Tuckerman et al. (1999), who use the notion of phase-space compressibility (see Chapter 17, Section 17.8, page 495), we shall show that the isokinetic thermostat produces a canonical distribution in conﬁguration space. Start with the deﬁnition of phase-space compressibility κ for non-Hamiltonian ﬂow (17.165): ˙ κ = ∇ · z.

(6.110)

Tuckerman et al. showed (see page 495) that, if a function w(z) can be deﬁned whose time derivative equals κ, then exp[−w(z)]dz1 , . . . , dz2n is an invariant volume element along the trajectory, meaning that exp[−w(z)] is

6.5 Controlling the system

199

the equilibrium weight function in phase space. Now κ follows from α: n ∂ p˙i = −nα, κ= ∂pi

(6.111)

i=1

where n is the number of degrees of freedom (= 3N − nc ). We can express α in phase-space functions by realizing that (in cartesian coordinates) pi p˙i d p2i = = 0, dt 2mi 2mi and therefore

n

n

i=1

i=1

n pi Fi i=1

Hence

since

mi

−α

n p2i = 0. mi

(6.112)

(6.113)

i=1

pi Fi pi Fi /mi dV (q) , = −β =β κ = −n i 2 m dt p /m i i i i i

(6.114)

Fi pi dV (q) ∂V = q˙i = − . dt ∂qi mi i

i

Thus the w = βV and the weight function equals exp[−βV ]. We conclude that ensemble averages of a variable A(z) over the isokinetic thermostat are given by A(z) exp[−βV (q)]δ( i p2i /2mi − nkB T ) dz . (6.115) A = exp[−βV (q)]δ( i p2i /2mi − nkB T ) dz Strong coupling can also be applied to constrain the pressure to a preset value (Evans and Morriss, 1983a; 1984). Combined with the isokinetic thermostat Evans and Morriss (1983b) have shown that a NPT ensemble (on the hypersurface of constrained pressure and kinetic energy) is obtained with a weight factor equal to exp[−βH], where H is the enthalpy U + pV . We do not pursue this type of pressure control further as it seems not to have become common practice, but refer instead to Allen and Tildesley (1987).

6.5.3 Weak-coupling methods Weak-coupling methods (Berendsen et al., 1984) are not stochastic in nature, and can be applied both for temperature and pressure control. For temperature control they do have the same eﬀect as a Langevin thermostat on the variance of velocities (i.e., on the temperature). The idea is to rescale

200

Molecular dynamics

velocities per step in such a way that the total temperature T of the system will decay with a ﬁrst-order process to the desired temperature T0 : dT T0 − T = . dt τ

(6.116)

This rate equation would cause a temperature deviation from T0 to decay exponentially to zero with a time constant τ . The implementation in terms of a scaling factor λ for the velocities is given by Δt T0 2 λ =1+ −1 , (6.117) τ T where T is given by the kinetic energy found after updating the velocities in a normal dynamic step. For the smallest possible value of time constant τ = Δt the scaling is complete and the temperature is exactly conserved. This corresponds to the Gauss isokinetic thermostat which produces a canonical ensemble in conﬁguration space. For τ much longer than the intrinsic correlation times for internal exchange of energy, the scaling has no eﬀect and a microcanonical ensemble is obtained. This is borne out by the ﬂuctuations in kinetic and potential energy: for small τ the kinetic energy does not ﬂuctuate but the potential energy does; as τ increases, ﬂuctuations in kinetic energy appear at the expense of potential energy ﬂuctuations, to become equal and opposite at large τ . The cross-over occurs roughly from a factor of 10 below to a factor of 10 above the intrinsic relaxation time for the exchange between kinetic and potential energy, which is system-dependent. Morishita (2000), in an attempt to characterize the weak-coupling ensemble, derived the following equation for the compressibility κ:

2 2 dV (q) −1 κ = β − β δK + O(N ) , (6.118) n dt plus some unknown function of p. Here δK is the ﬂuctuation of K, which depends on τ . For small τ , when δK = 0, this form reduces to βdV /dt, yielding the canonical conﬁguration space distribution derived above for the isokinetic thermostat. For large τ , when δK = −δV , the conﬁguration space distribution tends to

1 2 2 f (q) = exp −βV (q) − β (δV ) , (6.119) n which equals the microcanonical conﬁguration space distribution already derived earlier by Nos´e (1984a). For intermediate values of τ Morishita made the assumption that the ﬂuctuation of K is related to that of V by

6.5 Controlling the system

201

δK = −αδV , with α depending on τ , and arrives at a conﬁguration space distribution α f (q) = exp −βV (q) − β 2 (δV )2 . (6.120) n The distribution in momentum space remains unknown, but is less important as the integration over canonical momenta can be carried out separately. Note again that this is valid for cartesian coordinates only. Also note that the weak-coupling algorithm is no longer time-reversible, unless in the two extreme cases τ = Δt and τ → ∞. Pressure control by weak coupling is possible by scaling coordinates. In the spirit of weak coupling one attempts to regulate the pressure P according to the ﬁrst-order equation 1 dP = (P0 − P ). dt τp

(6.121)

Assume the isothermal compressibility βT = −(1/V )∂V /∂P is known, then scaling coordinates and volume, r = χr,

(6.122)

V = χ3 V,

(6.123)

every time step with a scaling factor χ, given by χ3 = 1 − βT

Δt (P0 − P ), τp

(6.124)

will accomplish that task. As the compressibility only enters the algorithm in conjunction with the time constant, its value need not be precisely known. The weak pressure coupling has the advantage of smooth response, but the disadvantages are that it does not generate a known ensemble and ﬂuctuations cannot be used.

6.5.4 Extended system dynamics The idea to extend the system with an extra degree of freedom that can be used to control a variable in the system, was introduced by Nos´e (1984a, 1984b) for temperature control. The method involved a somewhat inconvenient time scaling and was modiﬁed by Hoover (1985) into a scheme known as the Nos´e–Hoover thermostat, which has been widely used since. We treat the thermostat case, but the same principle can be applied to control pressure in a barostat. An extra variable η is introduced, which is a factor scaling the velocities. It has an associated “momentum” pη = Qη, ˙ where Q

202

Molecular dynamics

is the “mass” of the extra degree of freedom. The equations of motion are (p, q, m stand for all pi , qi , mi ): p , m pη p˙ = F (q) − p , Q p2 i p˙η = − nkB T. 2mi q˙ =

(6.125) (6.126) (6.127)

The temperature deviation from the bath temperature drives the time derivative of the velocity scaling factor, rather than the scaling factor itself, as is the case in the weak-coupling method. This makes the equations of motion time reversible again, and allows to compute the phase space distribution. On the other hand, it has the practical disadvantage that the temperature control is now a second-order diﬀerential equation in time, which leads to oscillatory approaches to equilibrium. Hoover shows that the equilibrium phase space distribution is given by

p2 1 2 i f (q, p, η, pη ) ∝ exp −β V (q) + , (6.128) + Qη 2mi 2 i

which is canonical. The extra variable is statistically independent of positions and velocities. The Nos´e–Hoover thermostat has been criticized because its behavior is non-ergodic (Toxvaerd and Olsen, 1990), which led Martyna et al. (1992) to formulation of the Nos´e–Hoover chain thermostat. In this thermostat there is a sequence of M additional variables η1 , η2 , . . . , ηM with their masses and conjugate momenta, each scaling its predecessor in the chain: q˙ = p˙ = η˙ 1 = p˙η1 = η˙ j = p˙ηj =

p , m pη F (q) − p 1 , Q1 pη1 , Q1 p2 pη i − nkB T − pη1 2 , 2mi Q2 pηj j = 2, . . . , M, Qj p2ηj−1 pη − kB T − pηj j+1 , Qj−1 Qj+1

(6.129) (6.130) (6.131) (6.132) (6.133) (6.134)

6.5 Controlling the system

203

1.4

T (ε/kB)

1.3

1.2

1.1

1

LD Berendsen NoséHoover 0

1

2

3

4

5

*

t

Figure 6.12 The temperature response of a Lennard–Jones ﬂuid under control of three thermostats (solid line: Langevin; dotted line: weak-coupling; dashed line: Nos´e–Hoover) after a step change in the reference temperature (Hess, 2002a, and by permission from van der Spoel et al., 2005.)

p˙ηM =

p2ηM −1 QM −1

− kB T.

(6.135)

6.5.5 Comparison of thermostats A comparison of the behavior in their approach to equilibrium of the Langevin, weak-coupling and Nos´e–Hoover thermostats has been made by Hess (2002a). Figure 6.12 shows that – as expected – the Nos´e–Hoover thermostat shows oscillatory behavior, while both the Langevin and weak-coupling thermostats proceed with a smooth exponential decay. The Nos´e–Hoover thermostat is therefore much less suitable to approach equilibrium, but it is more reliable to produce a canonical ensemble, once equilibrium has been reached. D’Alessandro et al. (2002) compared the Nos´e–Hoover, weak-coupling and Gaussian isokinetic thermostats for a system of butane molecules, covering a wide temperature range. They conclude that at low temperatures the Nos´e–Hoover thermostat cannot reproduce expected values of thermodynamic variables as internal energy and speciﬁc heat, while the isokinetic thermostat does. The weak-coupling thermostat reproduces averages quite well, but has no predictive power from its ﬂuctuations. These authors also monitored the Lyapunov exponent that is a measure of the rate at which trajectories deviate exponentially from each other; it is therefore an indica-

204

Molecular dynamics

tor of the tendency for chaotic behavior. A high Lyapunov exponent could be interpreted as a more eﬃcient sampling. It turns out that the isokinetic thermostat has the highest exponent, while the Nos´e–Hoover thermostat shows very low exponents. The weak coupling is in between. The stochastic thermostat was not studied, but is likely to show a high Lyapunov exponent as well.

6.6 Replica exchange method Molecular dynamics simulations are usually carried out at a given temperature, using some kind of thermostat, as described in the previous section. A representative initial conﬁguration is chosen and – after an initial equilibration period – one expects the system to reach thermal equilibrium. However, the system of interest may well have two or several potential wells, separated by relatively high barriers that are not eﬀectively crossed during the simulation time. This is the case for proteins and nucleic acids in solution that need a macroscopic time (say, seconds) to fold into a speciﬁc conformation, but also for large polymers, e.g., copolymer melts that need long rearrangement times to settle to a structure with minimum free energy. Also glasses below the glass transition temperature will “freeze” into a subset of the possible states. Such systems are not ergodic within the available simulation time; they will be trapped in a limited set of conﬁgurations (often called a conformation) that is a subset of the complete canonical distribution. How can one be sure that the simulated system is representative for the thermal equilibrium distribution? How can one prepare a proper initial state that samples a conformation with low free energy? Several methods have been devised in the past to overcome this problem. The most common approach is to use additional external information. For example, the dynamics of a folded protein can be studied by using a starting structure derived from experimental X-ray diﬀraction data; obviously the system is then forced into a predeﬁned conformation without any guarantee that this conformation is the lowest free-energy state compatible with the force ﬁeld used. Similarly, when crystallization does not occur spontaneously upon lowering the temperature in a simulation of a ﬂuid, one can start from an experimental crystal structure and study its equilibrium with the melt. A more satisfying approach is to start with a high-temperature simulation, which allows the frequent crossing of barriers, and let the system ﬁnd a low free-energy state by slowly lowering the temperature. This process is called tempering as it resembles the tempering of metals by slow cooling. Fast cooling will cause the system to become trapped in a low-

6.6 Replica exchange method

205

energy state, which may not at all be representative for a low free-energy conformation. However, by keeping the system at an elevated temperature just high enough to allow essential barrier crossings, the probability to end up in a low free-energy conformation increases. In metallurgy this annealing process leads to the annihilation of lattice defects. A similar computational process, called simulated annealing, was proposed in the 1980s in a seminal paper by Kirkpatrick et al. (1983) that has stimulated several computational innovations. A further development that has led to a breakthrough in the eﬃcient generation of a representative ensemble in cases where equilibration is slow, is now known by the name replica exchange method (REM). Both Monte Carlo and molecular dynamics versions are possible. In essence, an ensemble is generated that contains not only a number of conﬁgurations belonging to a canonical distribution at a given temperature, but also a set of temperatures. By including an exchange mechanism between diﬀerent temperatures, a total ensemble is generated that encompasses the whole set of temperatures. The eﬀect is a much faster relaxation than a single system would have at a low temperature: each system now rapidly visits a range of temperatures. Replica-exchange methods are ideally suited for parallel computers; each replica runs on a separate processor and there is only communication between processors when exchanges are attempted. The method is not restricted to a range of temperatures, but may also involve a range of Hamiltonians, e.g., with diﬀerent interaction parameters. In fact, the method was originally applied to Monte Carlo simulations of spin glasses, involving a range of values for the spin coupling constant (Swendsen and Wang, 1986). Since then there have been several developments, including multicanonical Monte Carlo (Berg and Neuhaus, 1991; Hansmann and Okamoto, 1993) and simulated tempering (Marinari and Parisi, 1992), but the most generally useful method with application to protein folding is the replica exchange molecular dynamics (REMD) method of Sugita and Okamoto (1999), which we now describe in a slightly simpliﬁed version.39 Consider M replicas S1 , . . . , SM of a system, subjected to canonical molecular dynamics (or Monte Carlo) simulations at M diﬀerent temperatures T1 , . . . , TM or inverse temperatures divided by kB , β1 , . . . , βM . Initially, system Si is at temperature Ti , but we allow exchange between the temperatures of two systems, so that in general system Si has temperature Tm . We order the temperatures always sequentially: m = 1, 2, . . . , M , but the sequence {i} = i(1), i(2), . . . , i(M ) of the systems is a permuta39

We use the canonical probability in conﬁguration space only, not in the full phase space.

206

Molecular dynamics

tion of the sequence 1, 2, . . . , M . A state X in the generalized ensemble consists of M conﬁgurations r i(1) , r i(2) , . . . , r i(M ) , with potential energies Ei(1) , Ei(2) , . . . , Ei(M ) . Because there are no interactions between the systems and each temperature occurs exactly once, the probability of this state is w ∝ exp[−(β1 Ei(1) + β2 Ei(2) + · · · + βM Ei(M ) )].

(6.136)

Now consider a possible exchange between two systems, e.g., systems Si at βm and Sj at βn . After the exchange, system Si will be at βn and Sj will be at βm . The probabilities before and after the exchange must be, respectively, wbefore ∝ exp[−(βm Ei + βn Ej )],

(6.137)

wafter ∝ exp[−(βn Ei + βm Ej )].

(6.138)

wafter = e−Δ , wbefore

(6.139)

Δ = (βn − βm )(Ei − Ej ).

(6.140)

and

Their ratio is

where

The transition probabilities W→ (meaning from “‘before” to “after”) and W← (meaning from “‘after” to “before”) must fulﬁll the detailed balance condition: wbefore W→ = wafter W← .

(6.141)

W→ = e−Δ . W←

(6.142)

Thus it follows that

This is accomplished by the Metropolis acceptance criterion: for for

Δ ≤ 0 : W→ = 1 −Δ

Δ > 0 : W→ = e

(6.143) ,

(6.144)

as is easily seen by considering the backward transition probability: for

Δ < 0 : W← = eΔ

(6.145)

for

Δ ≥ 0 : W← = 1,

(6.146)

which fulﬁlls (6.142). Although exchange between any pair may be attempted, in practice only neighboring temperatures yield non-zero acceptance ratios and the exchange

6.7 Applications of molecular dynamics

207

attempt can be limited to neighbors. An acceptance ratio of 20% is considered reasonable. One should choose the set of temperatures such that the acceptance ratio is more or less uniform over the full temperature range. This will depend on the system; Sugita and Okamoto (1999) ﬁnd for a peptide that an exponential distribution (equal ratios) is satisfactory. They use eight temperatures between 200 and 700 K, but a higher number (10 to 20) is recommended. The exchange, once accepted, can be accomplished in various ways. In Monte Carlo simulations, one exchanges both the random step sizes and the β’s in the acceptance criterion. In dynamics, the least disturbing implementation is to use a weak-coupling or a Langevin thermostat and switch the reference temperatures of the thermostats. The simulation should then extend over several time constants of the thermostat before another exchange is attempted. Using an isokinetic thermostat, the velocities should be scaled proportional to the square root of the temperature ratio upon exchange. In principle, it is also possible to switch not only the thermostats, but also all velocities of the particles between the two systems. This will drastically break up the time correlation of concerted motions; it is not clear whether this is advantageous or disadvantageous for the sampling eﬃciency.

6.7 Applications of molecular dynamics Molecular dynamics simulations with atomic detail can be routinely applied to systems containing up to a million particles over several nanoseconds. The time step for stable simulations is determined by the highest frequencies in the system; as a rule of thumb one may assume that at least ten, but preferably 50 time steps should be taken within the shortest period of oscillation.40 If the system contains mobile hydrogen atoms, bond oscillation periods may be as short as 10 fs; bond vibration involving heavy atoms typically exceed 20 fs. When covalent bonds are constrained, the highest frequencies are rotational and librational modes that involve hydrogen; dihedral angle rotations involving hydroxyl groups have similar periods. For example, in liquid water librational frequencies up to 800 cm−1 (40 fs) occur. A usual time step for simulations of systems containing liquid water is 1 to 2 fs when internal bond constraints are imposed; with further restrictions of hydrogen motions, “hydrogen-rich” systems as hydrated proteins remain stable with time steps up to 7 fs (Feenstra et al., 1999). In 2005, simulation of a typical medium-sized protein (lysozyme) in water, totalling some 30 000 40

See Berendsen and van Gunsteren (1981). Mazur (1997) concludes that even less than 10 steps per period suﬃce for the leap-frog algorithm.

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atoms, reached about 1 ns per day on a single state-of-the-art processor (van der Spoel et al., 2005), and this performance is expected to increase by a factor of ten every ﬁve years, according to Murphy’s law, which has been followed quite closely over the last decades. The availability of massively parallel clusters of processors allows simulation of much larger system sizes and much longer time scales. Proteins can be followed over microseconds, which is not yet suﬃcient to simulate realistic folding processes and reliably predict protein structures from sequence data. With the near future petaﬂop computers, the protein folding problem, which has been called the Holy Grail of biophysics (Berendsen, 1998), is likely to be solved. In material science, micron-size solids simulated for microseconds will become a reality. Figure 6.13 shows several snapshots of a simulation involving more than a billion (109 ) particles by Abraham et al. (2002).41 The simulated system is a crystal of Lennard–Jones particles, modeled to mimic a copper crystal, which is subjected to external tension forces that cause a crack to increase in size. The purpose of this simulation is to investigate the formation and propagation of dislocations that characterize the crack and model the process of work-hardening of metals. The system is a slab with 1008 atoms along the three orthogonal sides. Two notches are centered midway along the xdirection, at y = 0 and y = Ly , with a y-extension of 90 atomic layers which extends through the entire thickness Lz . The exposed notch faces are in the y − z planes with (110) faces, and the notch is pointed in the (1, −1, 0) direction. Periodic boundary conditions are imposed between the x−y faces at z = 0 and z = Lz . This notched slab geometry has a total of 1 023 103 872 atoms. The total simulation time for this study is 200 000 time-steps or 2 ns. The slab is initialized at zero temperature, and an outward strain of 4% is imposed on the outermost columns of atoms deﬁning the opposing vertical yz faces of the slab. The ﬁgures show only atoms with a potential energy less than 97 % of the bulk value magnitude. In the ﬁgures, we see a spaghetti-like network of atomic strings ﬂying from the vertices of the two opposing crack edges. This is simply a large number of mobile dislocations being created at each crack edge, rapidly ﬂowing through the stretched solid in an erratic manner, and eventually colliding with intersecting dislocations from the opposite edge. For the simple face-centered-cubic solid, dislocations are easily created at the apex of the two microcracks where the stress is at a maximum and easily ﬂow through 41

The author is indebted to Dr Farid Abraham for providing the pictures in Fig. 6.13 and the accompanying description. The interested reader is referred to Abraham (2003) for an introductory text on cracks and defects, and ductile and brittle behavior of metals.

Exercises

209

the solid giving rise to the ductility of the solid. The simulation supports the prevailing view that even though there may not be enough dislocations originally present in a crystal to account for the extensive slip in a ductile material (in this simulation there are initially no dislocations), their creation in vast amounts can occur at severe stress concentrations, such as at crack tips, enabling a stressed solid to be rapidly ﬁlled with dislocations and giving rise to material deformation under a steady load. The ﬁgures show snapshots of the propagating dislocations and rigid junctions evolving into a complex topology of the defect-solid landscape. Colliding dislocations can cause permanent atomic relocation along a line, called a rigid junction or sessile dislocation. A coarse-grain threedimensional skeleton of such sessile dislocations becomes apparent from a distant view. They are obstacles to further dislocation mobility. If their density is suﬃciently high, dislocation mobility becomes insigniﬁcant, and ductility of the solid ceases. The solid no longer can deform through dislocation motion: the ductile solid becomes brittle through this work-hardening process. Thus the simulation, with an impressive billion atoms still representing only a very small solid of 0.3 μm size, gives detailed insight into the mechanisms of work-hardening. The dynamical time span is on the order of a few nanoseconds, enough time for the phenomenon to achieve a ﬁnal structure state for the small size solid cube.

Exercises 6.1 6.2 6.3 6.4

Write out the operator product U(Δt) ≈ Uf (Δt/2)Uv (Δt)Uf (Δt/2) to obtain the velocity-Verlet algorithm. Obtain another algorithm by interchanging Uv and Uf . Solve (6.107). Compute (in reduced units) the period of oscillation of two Lennard– Jones particles at the distance where their interaction energy is minimal. What would be an appropriate time step (in reduced units) for a leap-frog simulation of a Lennard–Jones ﬂuid? How many fs is that time step for argon?

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Molecular dynamics

Figure 6.13 Five snapshots from a one-billion particle MD simulation of the propagation of a crack in a copper crystal under tension, showing the massive formation of dislocations. See text for details. Figures were kindly provided by Dr Farid Abraham of IBM Research and Lawrence Livermore National Laboratory, Livermore, CA, USA.

7 Free energy, entropy and potential of mean force

7.1 Introduction As we know from the applications of thermodynamics, free energy is much more important than energy, since it determines phase equilibria, such as melting and boiling points and the pressure of saturated vapors, and chemical equilibria such as solubilities, binding or dissociation constants and conformational changes. Unfortunately, it is generally much more diﬃcult to derive free energy diﬀerences from simulations than it is to derive energy diﬀerences. The reason for this is that free energy incorporates an entropic term −T S; entropy is given by an integral over phase space, while energy is an ensemble average. Only when the system is well localized in space (as a vibrating solid or a macromolecule with a well-deﬁned average structure) is it possible to approximate the multidimensional integral for a direct determination of entropy. This case will be considered in Section 7.2. Free energies of substates can be evaluated directly from completely equilibrated trajectories or ensembles that contain all accessible regions of conﬁgurational space. In practice it is hard to generate such complete ensembles when there are many low-lying states separated by barriers, but the ideal distribution may be approached by the replica exchange method (see Section 6.6). Once the conﬁgurational space has been subdivided into substates or conformations (possibly based on a cluster analysis of structures), the free energy of each substate is determined by the number of conﬁgurations observed in each substate. One may also observe the density of conﬁgurations along a deﬁned parameter (often called an order parameter or a reaction coordinate, which is a function of the coordinates) and derive the potential of mean force along that parameter. When a replica-exchange method has been used, the free energies of substates are obtained simultaneously for a range of temperatures, providing energies, entropies and speciﬁc heats. 211

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Free energy, entropy and potential of mean force

N

S

(2)

(2)

S´

(1)

(3)

(3)

N

(4)

S´

S E

(a)

(1)

(4)

E

(b)

Figure 7.1 Two thermodynamic cycles, allowing the replacement of one path by the sum of three other paths (see text). (a) Hydration of an ion (+) through the intermediate of a neutral atom (N), (b) binding of a substrate S to an enzyme E (dark gray), compared to the binding of another, similar, substrate S. Light gray squares represent an aqueous environment.

In general we cannot evaluate highly multidimensional integrals and all we have is a trajectory or ensemble in phase space or in conﬁgurational space. Then we must apply tricks to derive free energy diﬀerences from ensemble averages. This chapter is mostly about such tricks. In Section 7.3 Widom’s particle insertion method is treated, which relates the ensemble average of the Boltzmann factor of an inserted particle to its thermodynamic potential. Section 7.4 is about perturbation and integration methods that relate the ensemble average of the Boltzmann factor of a small perturbation to the diﬀerence in free energy, or the ensemble average of a Hamiltonian derivative to the derivative of the free energy. In Section 7.5 we ﬁrst make a clear distinction between two kinds of free energy: free energy of a thermodynamic state, and free energy in a restricted space as a function of one or more reaction coordinates. The latter is called a “potential of mean force,” or PMF, because its derivative is the ensemble-averaged force. The PMF is indispensable for simulations in reduced dimensionality (Chapter 8), as it provides the systematic forces in such cases. In Section 7.6 the connection between PMF and free energy is made and Section 7.7 lists a number of methods to determine potentials of mean force. Finally, Section 7.8 considers the determination of free energy diﬀerences from the work done in non-equilibrium pathways. Practical questions always concern free energy diﬀerences between thermodynamic states. So we need to obtain free energy diﬀerences from simulations. But, by the use of thermodynamic cycles, the computed pathways and

7.2 Free energy determination by spatial integration

213

intermediates may diﬀer from the experimental ones as long as the end results match. Thus one may choose even physically unrealistic or impossible intermediates to arrive at the required free energy diﬀerences. For example, if we wish to compute the free energy of hydration of a sodium ion, the interest lies in the diﬀerence in standard free energy ΔG01 of process (1) below.1 Here, (g) means ideal gas referred to standard concentration and (aq) means inﬁnitely dilute aqueous solution referred to standard concentration. For all processes, constant pressure and temperature are assumed. Now, process (1) can be built up from processes (2), (3) and (4), with a neutral, sodium-like atom N as intermediate (see Fig. 7.1a). Here N may be any convenient intermediate, e.g., a repulsive cavity with a size close to that of a sodium ion. (1) Na+ (g) → Na+ (aq) ΔG01 (2) N(g) → Na+ (g) ΔG02 (3) N(g) → N(aq) ΔG03 + (4) N(aq) → Na (aq) ΔG04 Going clockwise around the thermodynamic cycle, the total free energy change must be zero: ΔG01 − ΔG04 − ΔG03 + ΔG02 = 0.

(7.1)

Therefore, ΔG01 can be determined from three diﬀerent processes, which are each simpler and more eﬃcient to compute. The intermediate N can be chosen to optimize computational eﬃciency, but it can also be chosen to provide a single intermediate for the hydration of many diﬀerent ions. Process (1) in Fig. 7.1b represents the binding of a substrate S to an enzyme E. ΔG01 of this process yields the equilibrium binding constant KES . The direct determination of ΔG01 by simulation is diﬃcult, but the thermodynamic cycle allows to determine the binding constant of S relative to another similar substrate S, for which the binding constant KES of process (3) is known, by two simple processes (2) and (4).

7.2 Free energy determination by spatial integration Consider a system of N particles that are on the average situated at positions r i , i = 1, . . . , N , and ﬂuctuate randomly about those positions. Such systems are called non-diﬀusive when the mean-squared ﬂuctuations are ﬁnite 1

This is a simpliﬁed model; even process (1) is not a physically realistic process. It must be considered as part of a larger cycle involving a negative ion as well, as realistic thermodynamic experiments require electroneutral species.

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Free energy, entropy and potential of mean force

and stationary.2 Without loss of generality we may assume that the system as a whole has no translational and rotational motion and only consider the intra-system free energy. If the system is freely translating and rotating, as a macromolecule in dilute solution, there will be ideal-gas translational and rotational contributions to the total free energy (see Section 17.5.3 on page 468); the partition function (p.f.) is the product of a translational p.f (17.75), a rotational p.f. (17.82) and an internal partition function Qint . If the system is a non-translating, non-rotating crystal, the total free energy will consist of the free energy due to lattice vibrations and the internal free energy, assuming that the coupling between the two is negligible. What interests us here is the computation of the internal free energy, based on an ensemble of conﬁgurations in a system-ﬁxed cartesian coordinate system, obtained by a proper molecular dynamics or Monte Carlo procedure. Moreover, as the interest lies in the relative stability of diﬀerent conformations (or clusters of conﬁgurations), we are not interested in the absolute value of free energies or entropies. Kinetic contributions will cancel out in the diﬀerence between conformations. The determination of entropy directly from simulations was ﬁrst discussed by Karplus and Kushick (1981). Applications were published – among others – by Edholm et al. (1983) on the entropy of bilayer membranes and by DiNola et al.(1984) on the entropy diﬀerences between macromolecular conformations. The method has been evaluated with an error analysis by Edholm and Berendsen (1984) and the systematic errors on incomplete equilibration were discussed by Berendsen (1991a). A major improvement has been proposed by Schlitter (1993). The method has not become a standard: it is not applicable to diﬀusive systems, it cannot be easily applied to macromolecules in solution and considerable computational eﬀort is required for slowly relaxing systems. The essential contribution to the free energy that depends on a multidimensional integral is the conﬁgurational entropy: (7.2) Sconf = const − kB w(q) ln w(q) dq, where q is the set of generalized coordinates that exclude translational and rotational degrees of freedom (and also constrained degrees of freedom if applicable) and w(q) is the joint probability distribution for all the remaining n = 3N − 6 − nc coordinates. The constant in this equation comes from integration over the kinetic degrees of freedom (the conjugated momenta), 2

This is in contrast to diﬀusive systems, such as liquids, in which the mean-squared ﬂuctuations increase with – usually proportional to – time.

7.2 Free energy determination by spatial integration

215

but also contains a term due to the mass tensor (see page 401) that may depend on q. The inﬂuence of the mass tensor on conformational diﬀerences is often negligible and usually neglected or disregarded. Equation 7.2 is still an unsolvable multidimensional integral. But in nondiﬀusive systems it is possible to derive the most relevant information on the multidimensional distribution. For example, we can construct the correlation matrix of the coordinate ﬂuctuations: Cij = (qi − qi )(qj − qj ),

(7.3)

C = (Δq)(Δq)T ,

(7.4)

or where i, j = 1, . . . , n run over the generalized coordinates and Δq = q − q. It is generally not possible to assess higher correlations with any accuracy. If only the matrix of ﬂuctuations C is known, one can estimate the maximum entropy of any multidimensional distribution with a given C. By maximizing Sconf with respect to w under the conditions: w(q) dq = 1, (7.5) w(q)Δqi Δqj dq = Cij , (7.6) using Lagrange multipliers (see page 456), it appears that w(q) must be a multivariate Gaussian distribution in q: 1 w(q) = (2π)−n/2 (det C)−1/2 exp[− ΔqT C−1 Δq]. 2 The entropy of this distribution is

(7.7)

1 1 Smax = kB T n(1 + ln 2π) + kB T ln(det C). (7.8) 2 2 Thus, if the distribution really is a multivariate Gaussian, Smax is the conﬁgurational entropy; for any other distribution Smax is an upper bound for the entropy of the distribution: Sconf ≤ Smax .

(7.9)

The constants in (7.8) are irrelevant and all we need is the determinant of the correlation matrix of the positional ﬂuctuations. Generally it is possible to determine this entropy accurately; when equilibration is slow, the computed entropy tends to increase with the length of the simulation and approach the limit with a diﬀerence that is inversely proportional to the length of the simulation (DiNola et al., 1984; Berendsen, 1991).

216

Free energy, entropy and potential of mean force

It is also possible to derive the entropy from a principal component analysis of the positional ﬂuctuations in cartesian coordinates. In that case translational and rotational degrees of freedom must have been constrained, which is most easily done by subjecting all conﬁgurations to a standard translational–rotational ﬁt. The principal component analysis, often referred to as “essential dynamics” (Amadei et al., 1993), diagonalizes the correlation matrix of the positional ﬂuctuations, thus producing a new set of collective coordinates with uncorrelated ﬂuctuations. Each eigenvalue is proportional to the contribution of its corresponding collective degree of freedom to the total ﬂuctuation. There are 6 + nc zero (or very small) eigenvalues corresponding to the translation, rotation and internal constraints; these should be omitted from the entropy calculation. The determinant of C in (7.8) is now the product of all remaining eigenvalues. When there is more knowledge on the ﬂuctuations than the positional correlation matrix, the value of the entropy can be reﬁned. Each reﬁnement on the basis of additional information will decrease the computed entropy. For example, the marginal distributions wi (qi ) over single coordinates3 can usually be evaluated in more detail than just its variance. This is particularly important for dihedral angle distributions that have more than one maximum and that deviate signiﬁcantly from a Gaussian distribution. Dihedral angles in alkane chains have three populated ranges corresponding to trans, gauche − and gauche + conﬁgurations. It is then possible to compute the conﬁgurational entropy for each degree of freedom from (7.10) Smarg = −kB wi (qi ) ln wi (qi ) dqi and use that - after subtraction of the entropy of the marginal distribution had the latter been Gaussian with the same variance – as a correction to the entropy computed from the correlation matrix (Edholm and Berendsen, 1984). Another reﬁnement on the basis of extra knowledge is to exploit an observed clustering of conﬁgurations in conﬁgurational space: each cluster may be considered as a diﬀerent species and its entropy determined; the total entropy consists of a weighted average with an additional mixing term (see (7.55)). This determination of the classical entropy on the basis of positional ﬂuctuations has an important drawback: it computes in fact the classical entropy of a harmonic oscillator (h.o.), which is very wrong for high frequen3

The marginal distribution of qi is the full distribution integrated over all coordinates except qi .

7.2 Free energy determination by spatial integration

217

cies. For a one-dimensional classical h.o., the entropy is given by kB T , (7.11) ω which has the unfortunate property to become negative and even go to −∞ for large frequencies. Expressed in the variance x2 the classical entropy is 1 kB T ho 2 mx , (7.12) Scl = kB + kB ln 2 2 ho = kB + kB ln Scl

which becomes negative for small ﬂuctuations. This is entirely due to the neglect of quantum behavior. The eﬀect is unphysical, as even a constraint with no freedom to move – which should physically have a zero contribution to the entropy – has a negative, inﬁnite entropy. Equation (7.8) must be considered wrong: if any eigenvalue of the correlation matrix is zero, the determinant vanishes and no entropy calculation is possible. The correct quantum expression for the h.o. is kB ξ ho Squ , (7.13) = −kB ln 1 − e−ξ + ξ e −1 where ξ = ω/kB T . This can be expressed in terms of the classical variance, using mω 2 x2 = kB T , ξ= . (7.14) kB T mx2 The entropy now behaves as expected; it goes properly to zero when the ﬂuctuation goes to zero. In the multidimensional case one can use (7.13) after diagonalizing the correlation matrix of mass-weighted positional ﬂuctuations:

where

Cij = (Δxi )(Δxj ),

(7.15)

√ xi = xi mi .

(7.16)

Let us denote the eigenvalues of the C matrix by λk . Each eigenvalue corresponds to an independent harmonic mode with ξk = √

kB T λk

(7.17)

and each mode contributes the the total entropy according to (7.13). Now zero eigenvalues do not contribute to the entropy and do not cause the total entropy to diverge. For the exact quantum case one can no longer express the entropy in

218

Free energy, entropy and potential of mean force

terms of the determinant of the correlation matrix as in (7.8). However, by a clever invention of Schlitter (1993), there is an approximate, but good and eﬃcient, solution. Equation (7.13) is well approximated by S : e2 ho (7.18) Squ ≤ S = 0.5kB ln 1 + 2 , ξ yielding for the multidimensional case e2 kB T λk . S = 0.5kB ln Πk 1 + 2

(7.19)

Since the diagonal matrix λ is obtained from the mass-weighted correlation matrix C by an orthogonal transformation, leaving the determinant invariant, (7.19) can be rewritten as e2 kB T (7.20) C . S = 0.5kB ln det 1 + 2 This equation is a convenient and more accurate alternative to (7.8). Note that the mass-weighted positional ﬂuctuations are needed. In an interesting study, Sch¨ afer et al. (2000) have applied the Schlitter version of the entropy calculation to systems where the validity of a maximum-entropy approach based on covariances is doubtful, such as an ideal gas, a Lennard–Jones ﬂuid and a peptide in solution. As expected, the ideal gas results deviate appreciably from the exact value, but for the Lennard–Jones ﬂuid the computed entropy comes close (within ≈ 5%) to the real entropy. For a β-heptapeptide in methanol solution, which shows reversible folding in the simulations, the conﬁgurational entropy of the peptide itself can be calculated on the basis of the positional ﬂuctuations of the solute atoms. However, the contribution of the solvent that is missing in such calculations appears to be essential for a correct determination of free energy diﬀerences between conformational clusters. 7.3 Thermodynamic potentials and particle insertion Thermodynamic potentials μi of molecular species i are very important to relate microscopic to macroscopic quantities. Equilibrium constants of reactions, including phase equilibria, partition coeﬃcients of molecules between phases and binding and dissociation constants, are all expressed as changes in standard Gibbs free energies, which are composed of standard thermodynamic potentials of the participating molecules (see Section 16.7). How do thermodynamic potentials relate to free energies and potentials of mean force and how can they be computed from simulations?

7.3 Thermodynamic potentials and particle insertion

219

The thermodynamic potential of a molecular species is deﬁned as the derivative of the total free energy of a system of particles with respect to the number of moles ni of the considered species (see (16.8) on page 428): ∂G = NA {G(p, T, Ni + 1, Nj ) − G(p, T, Ni , Nj )}, (7.21) μi = ∂ni p,T,nj=i or, equivalently (see (16.27) on page 430), ∂A μi = ≈ NA {A(V, T, Ni + 1, Nj ) − A(V, T, Ni , Nj )}, (7.22) ∂ni V,T,nj=i where Ni is the number of particles of the i-th species, assumed to be large. The latter equation is the basis of the particle insertion method of Widom (1963) to determine the thermodynamic potential from simulations. The method is as ingenious as it is simple: place a ghost particle of species i in a random position and orientation in a simulated equilibrium conﬁguration. It is immaterial how the conﬁguration is obtained, but it is assumed that an equilibrium ensemble at a given temperature – and hence β – of conﬁgurations is available. Compute the ensemble average of the Boltzmann factor exp(−βVint ), where Vint is the interaction energy of the ghost particle with the real particles in the system. For a homogeneous system ensemble averaging includes averaging over space. The ghost particle senses the real particles, but does not interact with them and does not inﬂuence the ensemble. Now the thermodynamic potential is given by μexc = −RT lne−βVint ,

(7.23)

where μexc is the excess thermodynamic potential, i.e., the diﬀerence between the thermodynamic potential of species i in the simulated ensemble and the thermodynamic potential of species i in the ideal gas phase at the same density. This results follows from (7.22) and A = −kB T ln Q, yielding μ = −RT ln

Q(Ni + 1) . Q(Ni )

(7.24)

With (7.48) we can write dr ghost dr exp[−β{V (r) + Vint }] 2πmi kB T 3 1 Q(Ni + 1) , = Q(Ni ) h2 Ni + 1 dr exp[−βV (r)] (7.25) where r stands for the coordinates of all real particles and r ghost for the coordinates of the ghost particle. The ratio of integrals is the ensemble average of exp(−βVint ), integrated over the volume. But since – in a homogeneous

220

Free energy, entropy and potential of mean force

system – the ensemble average does not depend on the position of the ghost particle, this integration simply yields the volume V . Therefore:

2πmi kB T 3 V − RT lne−βVint . (7.26) μ = −RT ln h2 Ni + 1 The ﬁrst term is the thermodynamic potential of the i-th species in an ideal gas of non-interacting particles at a density (Ni + 1)/V . For large Ni this density is to a good approximation equal to Ni /V . Equation (7.26) is also valid for small Ni ; for example, when a water molecule is inserted into a ﬂuid consisting of another species (e.g., hexane), Ni = 0 and the excess thermodynamic potential is obtained with respect to the thermodynamic potential of ideal-gas water at a density of one molecule in the entire volume. For solutions we are interested in the standard thermodynamic potential and activity coeﬃcient of the solute and – in some cases – the solvent. How are these obtained from particle insertion? On the basis of molar concentration c, the thermodynamic potential (16.55) is expressed as γ c c μ(c) = μ0c + RT ln , (7.27) c0 where c0 is an agreed standard concentration (e.g., 1 molar) and μ0c is deﬁned by (16.58): c def μ0c = lim μ(c) − RT ln 0 . (7.28) c→0 c This deﬁnition guarantees that the activity coeﬃcient γc approaches the value 1 for inﬁnitely dilute solutions. A single measurement of μ(c) at one concentration can never determine both μ0c and γc . The standard thermodynamic potential requires a measurement at “inﬁnite” dilution. A single solute molecule in a pure solvent can be considered as inﬁnitely dilute since there is no interaction between solute particles. Thus, for inserting a single solute particle, the activity coeﬃcient will be unity and the thermodynamic potential is given by c μ = μ0c (solution) + RT ln 0 , (7.29) c where 1 . (7.30) c= NA V But since μ is also given by μ = μ(id.gas, c) + μexc c = μ(id.gas, c0 ) + RT ln 0 + μexc , c

(7.31) (7.32)

7.4 Free energy by perturbation and integration

221

it follows that μ0c (solution) = μ0c (id.gas) + μexc .

(7.33)

Here, μexc is “measured” by particle insertion according to (7.23). The standard concentration for the solution and the ideal gas must be the same. Note Thus far we have treated particle insertion into a canonical (N, V, T ) ensemble, which yields a relation between the averaged Boltzmann factor and the Helmholtz free energy A, based on (7.22). This equationis not exact and not valid for small numbers Ni . However, (7.21) is exact, as G = ni μi under conditions of constant pressure and temperature (see (16.12) on page 428); this relation is valid because both p and T are intensive quantities. Such a relation does not exist for A. It is possible to make corrections to A, using the compressibility, but it is more elegant to use a (N, p, T ) ensemble. The N, p, T average of the Boltzmann factor yields the thermodynamic potential exactly (see (17.33) and (17.31) on page 461).

The problem with the particle-insertion method is that in realistic dense ﬂuids the insertion in a random position nearly always results in a high, repulsive, interaction energy and hence in a negligible Boltzmann factor. Even with computational tricks that avoid the full computation of all interaction energies it is very diﬃcult to obtain good statistics on the ensemble average. A way out is to insert a smaller particle and – in a second step – let the particle grow to its full size and determine the change in free energy by thermodynamic integration (see next section). In cases where the diﬀerence in thermodynamic standard potential between two coexisting phases is required – as for the determination of partition coeﬃcients – a suitable method is to determine the potential of mean force over a path that leads from one phase into the other. The diﬀerence between the two plateau levels of the PMF in the two phases is also the diﬀerence in standard thermodynamic potential.

7.4 Free energy by perturbation and integration Consider a potential function V (r, λ) with a parametric dependence on a coupling parameter 0 ≤ λ ≤ 1 that modiﬁes the interaction. The two extremes, λ = 0 and λ = 1, correspond to two diﬀerent systems, A and B, respectively, with interaction functions VA (r) and VB (r): VA (r) = V (r, λ = 0),

(7.34)

VB (r) = V (r, λ = 1).

(7.35)

For example, system A may consist of two neon atoms dissolved in 1000 water molecules, while system B consists of one sodium ion and one ﬂuoride

222

Free energy, entropy and potential of mean force

ion dissolved in 1000 water molecules. The parameter λ changes the neon– water interaction into an ion–water interaction, essentially switching on the Coulomb interactions. Or, system A may correspond to a protein with a bound ligand LA in solution, while system B corresponds to the same protein in solution, but with a slightly modiﬁed ligand LB . The end states A and B represent real physical systems, but intermediate states with a λ unequal to either zero or one are artiﬁcial constructs. The dependence on λ is not prescribed; it can be a simple linear relation like V (λ) = (1 − λ)VA + λVB ,

(7.36)

or have a complex non-linear form. The essential features are that the potential is a continuous function of λ that satisﬁes (7.34) and (7.35). Now consider the Helmholtz free energy of the system at a given value of λ:

−βV (r ,λ) A(λ) = −kB T ln c e dr . (7.37) It is impossible to compute this multidimensional integral from simulations. But it is possible to compute A(λ + Δλ) as a perturbation from an ensemble average:4 exp[−βV (r, λ + Δλ)] dr (7.38) A(λ + Δλ) − A(λ) = −kB T ln exp[−βV (r, λ)] dr = −kB T ln e−β[V (λ+Δλ)−V (λ)] . (7.39) λ

It is also possible to compute dA/dλ from an ensemble average: ∂V ∂V dA ∂λ(r, λ) exp[−βV (r, λ)], dr = = . dλ ∂λ λ exp[−βV (r, λ)], dr

(7.40)

The averages must be taken over an equilibrium ensemble using V (λ). Thus, if the λ-path from 0 to 1 is constructed from a number of intermediate points, then the total ΔA = AB − AA = A(λ = 1) − A(λ = 0) can be reconstructed from the ensemble averages at the intermediate points. In general the most convenient and accurate reconstruction is from the derivatives at intermediate points by integration with an appropriate numerical procedure (Press et al., 1993), e.g., by computing a cubic spline with the given derivatives (see 4

This equation was ﬁrst given by Torrie and Valleau (1974) in the context of Monte Carlo simulations of Lennard–Jones ﬂuids. Pearlman and Kollman (1989a) have reﬁned windowing techniques for thermodynamic integration.

7.4 Free energy by perturbation and integration

223

Chapter 19).5 This procedure to ﬁnd diﬀerences in free energies is called thermodynamic integration. Integration can also be accomplished from a series of perturbations, using (7.39); in that case there may be systematic deviations if the interval is not very small and it is recommended to apply both positive and negative perturbations from each point and check the closure. The derivatives or perturbations are only reliable when the ensemble, over which the derivative or perturbation is averaged, is a proper equilibrium ensemble. In slowly relaxing systems there may be remains of history from the previous λ-point and the integration may develop a systematic deviation. It is recommended to perform the thermodynamic integration from both directions, i.e., changing λ from 0 to 1 as well as from 1 to 0. Systematic deviations due to insuﬃcient equilibration are expected to have opposite signs in both cases. So the obtained hysteresis is an indication of the equilibration error. A limiting case of thermodynamic integration is the slow-growth method, in which λ is changed with a small increment (Δλ)i at the i-th step in a molecular dynamics simulation, starting at 0 and ending at 1 (or vice versa).6 This increment may or may not be taken as a constant. Then the total change in free energy is approximated by ∂V (λ) ΔA = (Δλ)i . (7.41) ∂λ i i

Only in the limit of an inﬁnitely slow change of λ a true free energy diﬀerence will be obtained; if the growth is too rapid, the ensemble will “lag behind” the proper equilibrium at the actual value of λ. This will lead to an easily detected hysteresis when slow-growth simulations in forward and backward directions are compared. The average between the results of a forward and backward integration is always more accurate than either value. Figure 7.2 gives an early example of hysteresis in a simulation that changes a model neon atom into a sodium ion in aqueous solution by charging the atom proportional to time (Straatsma and Berendsen, 1988). In this case the free energy appears to change quadratically with time and the ensemble appears to relax quickly. Pearlman and Kollman (1989b) and Wood (1991) have 5

6

The standard error in the integral can be evaluated if the standard error σi in each ensemble average Ai is known. Numerical integration yields an integral that can be expressed as ΔA = wi (Ai ± σi ), where wi are weights depending on the interval and the procedure used. The ) wi2 σi2 . standard error in ΔA equals The slow-growth method was pioneered by Postma (1985), see Berendsen et al. (1985), and applied – among others – by Straatsma and Berendsen (1988) and by Pearlman and Kollman (1989b).

224

Free energy, entropy and potential of mean force ± ΔG (kJ/mol)

440

420

400

20

40

60 80 Total simulation time T (ps)

Figure 7.2 Free energy change resulting from transforming a model neon atom into a sodium ion in a bath of 216 water molecules, by linearly increasing the charge on the atom in a total simulation time T . Upward triangles are for growth from Ne to Na+ , yielding a negative ΔG; downward triangles are for the opposite change. Dashed curves are predictions from the theory of Wood (1991) assuming a relaxation time for the ensemble of 0.15 ps. Data are from Straatsma and Berendsen (1988).

analyzed the eﬀects of the rate of slow-growth free energy determinations. At present slow growth is used less frequently than integration based on a number of well-equilibrated intermediate points because the latter allows a better evaluation and optimization of the overall accuracy. Note A similar remark as was made in connection with particle insertion can be made here as well. In most applications one is interested in constant pressure rather than constant volume conditions. For example, in order to ﬁnd equilibrium constants, one needs ΔG0 . Using N, V, T ensembles one may need corrections to connect the end points at constant pressure rather than constant volume. It is much more elegant to use N, p, T ensembles, with partition function Δ (see (17.31) on page 461) that relate directly to Gibbs free energies. Ensemble averages of Hamiltonian derivatives now yield derivatives of G rather than A.

There are several practical considerations concerning the method used for the integration of free energy from initial to ﬁnal state. As computational integration is not limited to physically realistic systems (i.e., as long as the initial and ﬁnal states are realistic), there is almost no bound to the phantasy that can be introduced into the methodology. The word “computational

7.4 Free energy by perturbation and integration

225

alchemy” is not misplaced, as one may choose to change lead into gold, be it that the gold must – unfortunately – be returned to lead before a realistic result is obtained. We list a few tricks and warnings. • The free energy as a function of λ may not be well-behaved, so that numerical integration from a limited number of points becomes inaccurate. The density of points in the range 0 ≤ λ ≤ 1 can be chosen to optimize the integration accuracy, but ideally A(λ) should be a smooth function without steep derivatives, being well-represented by a polynomial of low order. One can manipulate the function by changing the functional dependence of the Hamiltonian on λ. • Ideally the free energy curve should be monotonous; if a large intermediate maximum or minimum occurs, computational eﬀort must be spent to compute compensating free-energy changes. A maximum may easily occur when there are highly repulsive conﬁgurations at intermediate values of λ’s. Such repulsive intermediates can be avoided by choosing appropriately smoothed potentials. • Replacing diverging functions as the repulsive r−12 or dispersion and Coulomb interactions by soft-core interactions for intermediate λ’s removes the singularities and allows particles to move through each other rather than having to avoid each other.7 The GROMACS software (van der Spoel et al., 2005) uses a modiﬁcation of the distance between particles of the form V (r) = (1 − λ)VA (rA ) + λVB (rB ), 2

6 1/6

rA = (cλ + r )

,

rB = [c(1 − λ)2 + r6 ]1/6 ,

(7.42) (7.43) (7.44)

while Tappura et al. (2000) switch to a function ar6 + b below a speciﬁed (short) distance, with a and b such that the potential function and its derivative are continuous at the switch distance. • Another non-physical intervention is to allow particles to move into a fourth spatial dimension for intermediate λ’s (van Gunsteren et al., 1993). Since there is much more space in four than in three dimensions, particles can easily avoid repulsive conﬁgurations. But they also loose their structural coherence and the motion in the fourth dimension must be carefully restrained. The method is more aesthetically appealing than it is practical. 7

Soft-core potentials were originally developed for structure optimization and protein folding (Levitt, 1983; Huber et al., 1997; Tappura et al., 2000).

226

Free energy, entropy and potential of mean force

• There can be problems when particles “vanish” at either end point of the integration path. When the interactions of an atom with its environment are made to vanish, the particle is still there in the simulation as an ideal gas atom. It has mass and velocity, but is uncoupled to other degrees of freedom and therefore does not equilibrate properly. Problems are avoided by constraining the vanishing particle to a ﬁxed position where it has neither a kinetic energy nor a conﬁgurational entropy and does not contribute to the free energy. One should take care that the λ-dependence is not diverging near the value of λ where the particle vanishes. Strictly speaking, one should also correct for the vanishing kinetic term (2πmkB T /h2 )−1/2 in the free energy, but that term will always be compensated when a complete, physically realistic, cycle is completed. • When particles are changed into other particles with diﬀerent mass, the free energy change has a diﬀerent kinetic term. It is possible to change the masses also with a coupling parameter, but there is no need to do that, as – just as in the case of a vanishing particle – the kinetic eﬀect will always be compensated when a complete, physically realistic, cycle is completed. Real free energy diﬀerences always concern the same number and type of particles on both sides of the reaction. • Be careful when the coupling parameter involves a constraint.8 For example, if one wishes to change a hydrogen atom in benzene into a methyl group (changing benzene into toluene), the carbon–particle distance will change from 0.110 to 0.152 nm. In a simulation with bond constraints, the constraint length is modiﬁed as a function of the coupling parameter. Each length modiﬁcation in the presence of a constraint force involves a change in free energy, as work is done against (or with) the constraint force. So the work done by the constraint force must be monitored. The constraint force Fc follows from the procedure used to reset the constraints (see Section 15.8 on page 417); if the constraint distance rc is changed by a small increment Δrc = (drc /dλ) Δλ, the energy increases with Fc Δrc . Thus there is a contribution from every bond length constraint to the ensemble average of ∂V /∂λ: ∂V drc = Fc . (7.45) ∂λ constr dλ In addition, there may be a contribution from the Jacobian of the transformation from cartesian to generalized coordinates, or – equivalently – from the mass-metric tensor (see Section 17.9.3 and speciﬁcally (17.199) on page 501). The extra weight factor |Z|−1/2 in the constrained ensemble 8

See van Gunsteren et al. (1993), pp 335–40.

7.5 Free energy and potentials of mean force

may well be a function of λ and contribute a term in dA/dλ: dA 1 ∂|Z| = kB T |Z|1/2 . dλ metric 2 ∂λ

227

(7.46)

The same arguments that are given in Section 17.9.3 to show that the metric eﬀects of constraints are often negligible (see page 502) are also valid for its λ-dependence. Even more so: in closed thermodynamic cycles the eﬀect may cancel. • A large improvement of the eﬃciency to compute free energy changes for many diﬀerent end states (e.g., ﬁnding the binding constants to a protein for many compounds) can be obtained by using a soft intermediate (Liu et al., 1996; Oostenbrink and van Gunsteren, 2003). Such an intermediate compound does not have to be physically realistic, but should be constructed such that it covers a broad part of conﬁgurational space and allows overlap with the many real compounds one is interested in. If well chosen, the change from this intermediate to the real compound may consist of a single perturbation step only.

7.5 Free energy and potentials of mean force In this section the potential of mean force (PMF) will be deﬁned and a few remarks will be made on the relations between PMF, free energy, and chemical potential. The potential of mean force is a free energy with respect to certain deﬁned variables, which are functions of the particle coordinates, and which are in general indicated as reaction coordinates because they are often applied to describe reactions or transitions between diﬀerent potential wells. What does that exactly mean, and what is the diﬀerence between a free energy and a potential of mean force? What is the relation of both to the chemical potential? The potential energy as a function of all coordinates, often referred to as the energy landscape, has one global minimum, but can have a very complex structure with multiple local minima, separated by barriers of various heights. If the system has ergodic behavior, it visits in an equilibrium state at temperature T all regions of conﬁguration space that have an energy within a range of the order of kB T with respect to the global minimum. It is generally assumed that realistic systems with non-idealized potentials in principle have ergodic behavior,9 but whether all relevant regions of conﬁg9

Idealized systems may well be non-ergodic, e.g., an isolated system of coupled harmonic oscillators will remain forever in the combination of eigenstates that make up its initial conﬁguration and velocities; it will never undergo any transitions to originally unoccupied eigenstates unless there are external disturbances or non-harmonic terms in the interaction function.

228

Free energy, entropy and potential of mean force V mf

P

R ?k T B 6

reaction coordinate ξ Figure 7.3 Potential of mean force in one “reaction coordinate” ξ. There are two regions of conﬁgurational space (R and P) that can be designated as conﬁning a thermodynamic state.

uration space will indeed be accessed in the course of the observation time is another matter. If the barriers between diﬀerent local minima are large, and the observation time is limited, the system may easily remain trapped in certain regions of conﬁguration space. Notable examples are metastable systems (as a mixture of hydrogen and oxygen), or polymers below the melting temperature or glasses below the glass transition temperature. Thus ergodicity becomes an academic problem, and the thermodynamic state of the system is deﬁned by the region of conﬁgurational space actually visited in the observation time considered. Consider a system that can undergo a slow reversible reaction, and in the observation time is either in the reactant state R or in the product state P. In the complete multidimensional energy landscape there are two local minima, one for the R and one for the P state, separated by a barrier large enough to observe each state as metastable equilibrium. Let the local minima be given by the potentials V0R and V0P . Then the Helmholtz free energy A (see Chapter 17) is given – for classical systems in cartesian coordinates – by

A = −kB T ln Q,

(7.47)

7.5 Free energy and potentials of mean force

with

e−βV (r ) dr,

Q=c

229

(7.48)

where the integration is carried out for all particle coordinates over all space. The constant c derives from integration over momenta, which for N particles consisting of species s of Ns indistinguishable particles (with mass ms ) equals c=

2πkB T h2

3N/2

3N /2

Πs

ms s . Ns !

(7.49)

Expressed in de Broglie wavelengths Λs for species s: Λs = √

h , 2πms kB T

(7.50)

and using Stirling’s approximation for Ns !, the constant c becomes Ns e . c = Πs Ns Λ3s

(7.51)

Note that c has the dimension of an inverse volume in 3N -dimensional space V −N , and the integral in (7.48) has the dimension of a volume to the power N . Thus, taking logarithms, we cannot split Q in c and an integral without loosing the metric independence of the parts. It is irrelevant what zero point is chosen to express the potential; addition of an arbitrary value V0 to the potential will result in multiplying Q with a factor exp(−βV0 ), and adding V0 to A. When each of the states R and P have long life times, and have local ergodic behavior, they can be considered as separate thermodynamic states, with Helmholtz free energies R R R A = −kB T ln Q Q =c e−βV (r ) dr (7.52) R QP = c e−βV (r ) dr, (7.53) AP = −kB T ln QP P

where the integrations are now carried out over the parts of conﬁguration space deﬁned as the R and P regions, respectively. We may assume that these regions encompass all local minima and that the integration over space outside the R and P regions does not contribute signiﬁcantly to the overall Q. We immediately see that, although Q = QR + QP , A = AR + AP . Instead, deﬁning the relative probabilities to be in the R and P state, respectively,

230

Free energy, entropy and potential of mean force

as wR and wP : wR =

QR QP and wQ = , Q Q

(7.54)

it is straightforward to show that A = wR AR + wP AP + kB T (wR ln wR + wP ln wP ).

(7.55)

The latter term is due to the mixing entropy resulting from the distribution of the system over two states. Note that the zero point for the energy must be the same for both R and P. Now deﬁne a reaction coordinate ξ(r) as a function of particle coordinates, chosen in such a way that it connects the R and P regions of conﬁguration space. There are many choices, and in general ξ may be a complicated nonlinear function of coordinates. For example, a reaction coordinate that will describe the transfer of a proton over a hydrogen bond X-H· · ·Y may be deﬁned as ξ = rXH /rXY ; ξ will encompass the R state around a value of 0.3 and the P state around 0.7. One may also choose several reaction coordinates that make up a reduced conﬁguration space; thus ξ becomes a multidimensional vector. Only in rare cases can we deﬁne the relevant degrees of freedom as a subset of cartesian particle coordinates. We ﬁrst separate integration over the reaction coordinate from the integral in Q: Q = c dξ dre−βV (r ) δ(ξ(r) − ξ). (7.56) Here ξ(r) is a function of r deﬁning the reaction coordinate, while ξ is a value of the reaction coordinate (here the integration variable).10 In the case of multidimensional reaction coordinate spaces, the delta-function should be replaced by a product of delta-functions for each of the reaction coordinates. Now deﬁne the potential of mean force V mf (ξ) as

def mf −βV (r ) V (ξ) = −kB T ln c dre δ(ξ(r) − ξ) , (7.57) so that

Q=

and

A = −kB T ln

10

e−βV

mf (ξ)

dξ,

−βV mf (ξ)

e

(7.58)

dξ .

(7.59)

Use of the same notation ξ for both the function and the variable gives no confusion as long as we write the function explicitly with its argument.

7.6 Reconstruction of free energy from PMF

231

Note that the potential of mean force is an integral over multidimensional hyperspace. Note also that the integral in (7.57) is not dimensionless and therefore the PMF depends on the choice of the unit of length. After integration, as in (7.58), this dependency vanishes again. Such inconsistencies can be avoided by scaling both components with respect to a standard multidimensional volume, but we rather omit such complicating factors and always keep in mind that the absolute value of PMFs have no meaning without specifying the underlying metric. It is generally not possible to evaluate such integrals from simulations. The only tractable cases are homogeneous distributions (ideal gases) and distribution functions that can be approximated by (multivariate) Gaussian distributions (harmonic potentials). As we shall see, however, it will be possible to evaluate derivatives of V mf from ensemble averages. Therefore, we shall be able to compute V mf by integration over multiple simulation results, up to an unknown additive constant.

7.6 Reconstruction of free energy from PMF Once the PMF is known, the Helmholtz free energy of a thermodynamic state can be computed from (7.59) by integration over the relevant part of the reaction coordinate. Thus the PMF is a free energy for the system excluding the reaction coordinates as degrees of freedom. In the following we consider a few practical examples: the harmonic case, both one- and multidimensional and including quantum eﬀects; reconstruction from observed probability densities with dihedral angle distributions as example; the PMF between two particles in a liquid and its relation to the pair distribution function; the relation between the partition coeﬃcient of a solute in two immiscible liquids to the PMF.

7.6.1 Harmonic wells Consider the simple example of a PMF that is quadratic in the (single) reaction coordinate in the region of interest, e.g in the reactant region R (as sketched in Fig. 7.3): 1 (7.60) V mf ≈ V0mf + k R ξ 2 . 2 Then the Helmholtz free energy of the reactant state is given by integration

1 kR R −βV mf (ξ) e dξ = V0mf + kB T ln . (7.61) A ≈ −kB T ln 2 2πkB T −∞+∞

232

Free energy, entropy and potential of mean force

Beware that the term under the logarithm is not dimensionless, but that the metric dependence is compensated in V0mf . We see that A becomes lower when the force constant decreases; the potential well then is broader and the entropy increases. In the multidimensional harmonic case the PMF is given by a quadratic term involving a symmetric matrix KR of force constants, which is equal to the Hessian of the potential well, i.e., the matrix of second derivatives: V mf ≈ V0mf + 12 ξ T KR ξ.

(7.62)

Integration according to (7.59) now involves ﬁrst an orthogonal transformation to diagonalize the matrix, which yields a product of one-dimensional integrals; carrying out the integrations yields a product of eigenvalues of the matrix, which equals the determinant of the diagonalized matrix. But the determinant of a matrix does not change under orthogonal transformations and we obtain 1 det KR . (7.63) AR ≈ V0mf + kB T ln 2 2πkB T Thus far we have considered the system to behave classically. However, we know that particles in harmonic wells (especially protons!), as they occur in molecular systems at ordinary temperature, are not at all close to the classical limit and often even reside in the quantum ground state. The classical expressions for the free energy are very wrong in such cases. The PMF well itself is generally determined from simulations or computations with constrained reaction coordinates in which the quantum character of the motion in the reaction coordinate does not appear. It is therefore relevant to ask what quantum eﬀects can be expected in the reconstruction of free energies from harmonic PMF wells. Quantum corrections to harmonic oscillator free energies can be easily made, if the frequencies of the normal modes are known (see Chapter 3, Section 3.5.4 on page 74). The problem with PMFs is that they do not represent pure Hamiltonian potentials in which particles move, and since the reaction coordinates are generalized coordinates which are (in general) non-linear functions of the particle coordinates, the eﬀective masses (or a mass tensor in multidimensional cases) are complicated functions of the coordinates. Instead of computing such eﬀective masses, the frequencies of the normal modes can much more easily be determined from a relatively short MD run with full detail in the potential well. Monitoring and Fourier˙ transforming the velocities ξ(t) of the reaction coordinates will reveal the eigenfrequencies of the motion of the reaction coordinates in the well of the

7.6 Reconstruction of free energy from PMF

233

βA

8 βAqu

6

βAqu

- Acl

4 2

βAcl

0 -2 -4

0.5

1

1.5

2

2.5

3

3.5

4 βhν

Figure 7.4 Helmholtz free energies divided by kB T for a single harmonic oscillator, as a function of hν/kB T , for both classical and quantum-mechanical statistics. The drawn line gives the quantum correction to a classical free energy.

PMF without bothering about the eﬀective masses of the resulting motion. There are as many eigenfrequencies (but they may be degenerate) as there are independent reaction coordinates. According to quantum statistics (see Chapter 17), each eigenfrequency ν leads to a contribution to the free energy of

1 1 qu Aν = kB T ln sinh βhν , (7.64) 2 2 which is to be compared to the classical contribution Acl ν = kB T ln(βhν).

(7.65)

One may use the diﬀerence to correct the classical free energy determination, and – from temperature derivatives – the enthalpy and entropy (see Fig. 7.4).

234

Free energy, entropy and potential of mean force

7.7 Methods to derive the potential of mean force In general a potential of mean force V mf (r ) describes the eﬀective potential that determines the motion of coordinates r in a reduced system, averaged over an equilibrium ensemble of the other coordinates r . In Chapter 8 the use of potentials of mean force in reduced systems is treated in detail. For simplicity we write r , r as a subdivision of cartesian space, but often the reduced system is described by a set of generalized coordinates. In this section we look at methods to derive potentials of mean force, which may then be useful for implementation in the reduced systems dynamics of Chapter 8. In most cases of interest (containing improbable areas of the primed space) it is impossible to determine V mf directly from an equilibrium simulation of the whole system. If it were, there would not be much point in reducing the number of degrees of freedom in the ﬁrst place. The following possibilities are open to derive a suitable potential of mean force: • From a macroscopic (generally a mean-ﬁeld) theory. For example, if we wish to treat a solvent as “irrelevant,” its inﬂuence on the electrostatic interactions of charges within the “relevant” particles and on the electrostatic contribution to the solvation free energy of (partially) charged particles, can be computed from electrostatic continuum theory (see Section 13.7). This requires solving the Poisson equation (or the Poisson– Boltzmann equation) with a ﬁnite-diﬀerence or Fourier method on a grid or with a boundary-element method on a triangulated surface. A computationally less demanding approximation is the generalized Born model (see Section 13.7.5 on page 351). Since such a treatment cannot be accurate on the atomic scale and misses non-electrostatic contributions, the electrostatic potential of mean force must be augmented by local interaction terms depending on the chemical nature and the surface accessibility of the primed particles. Another example is the treatment of all particles outside a deﬁned boundary as “irrelevant”. If the boundary of the primed system is taken to be spherical, the electrostatic terms may be represented by a reaction ﬁeld that is much simpler to compute than the Poisson equation for an irregular surface (see Section 13.7.4). • By thermodynamic integration. It is possible to obtain the derivative(s) of the potential of mean force at a given conﬁguration of r by performing a constrained equilibrium simulation of the full system and averaging the constraint forces (which are easily obtained from the simulation) over the double-primed ensemble. By performing a suﬃcient number of such simulations at strategically chosen conﬁgurations of the primed particles,

7.7 Methods to derive the potential of mean force

235

the potential of mean force can be obtained from numerical integration of the average constraint forces. This method is only feasible for a few (one to three) dimensions of the primed degrees of freedom, because the number of points, and hence full simulations, that is needed to reconstruct a V mf surface in n dimensions increases with the number of points in one dimension (say, 10) to the power n. By taking the gradient of (8.11), we ﬁnd that ∂V (r ,r ) −βV (r ,r ) e dr ∂V mf (r ) ∂ r i = ∂r i e−βV (r r ) dr ∂V (r , r ) = ∂r i = F ci . (7.66) The second line in the above equation gives – except for the sign – the average over the constrained ensemble of the internal force acting on the i-th primed particle; in a constrained simulation this force is exactly balanced by the constraint force F ci on that particle. These equations are modiﬁed for generalized coordinates (see den Otter and Briels, 1998; Sprik and Ciccotti, 1998; den Otter, 2000). • By thermodynamic perturbation. Instead of averaging the derivatives of the potential, we may also average the Boltzmann factor of a (small but ﬁnite) perturbation: −βV (r +Δr ,r ) e dr mf mf V (r + Δr ) − V (r ) = −kB T ln −βV (r ,r ) dr e = −k T ln e−β[V (r +Δr ,r )−V (r ,r )] . (7.67) B

This equation is exact, but statistically only accurate for small displacements. By choosing a suﬃciently dense net of conﬁgurations to generate the ensembles, the potentials of mean force can be reconstructed by ﬁtting perturbations of one point to those of a nearby point. • By umbrella sampling. This method, pioneered by Torrie and Valleau (1977), restrains, rather than constrains, the primed coordinates around a given conﬁguration by adding a restraining potential V u (r ) to the potential V (r , r ). This umbrella potential could, for example, be harmonic in shape. The resulting canonical umbrella distribution wu (r ) in the primed coordinates will in equilibrium be given by U wu (r ) ∝ dr e−βV (r ,r )−βV (r ) . (7.68)

236

Free energy, entropy and potential of mean force

Therefore, u V mf (r ) = constant − kT ln wu (r )e+βV (r ) = constant − kT ln[wu (r )] − V u (r),

(7.69)

which says that the potential of mean force can be reconstructed in the neighborhood of the restrained conﬁguration by keeping track of the distribution over the primed coordinates and correcting the bias caused by the umbrella potential. This reconstruction is only accurate in a region where suﬃcient statistics is obtained to determine wu (r ) accurately. The full potential of mean force can again be reconstructed by ﬁtting adjacent umbrella distributions to each other. An alternative to reconstructing the local V mf from the distribution function is averaging of the umbrella force, which is easily monitored in a MD simulation.11 In the case of a harmonic umbrella, the average force is also equal to the mean displacement of the coordinate(s) on which the umbrella is imposed (with respect to the umbrella center), divided by the harmonic force constant. The average umbrella force is approximately, but not exactly, equal to the derivative of the potential of mean force at the umbrella center. In fact, it is given exactly by a weighted average of the derivative over the umbrella distribution wu (r ): u dr w (r )[∂V mf (r )/∂r i ] u , (7.70) F u = dr wu (r ) which is accurate to second order (i.e., including the second derivative of the potential) to the derivative at the average position of the primed coordinate in the umbrella ensemble. This average can also be used to reconstruct the potential of mean force. Proof The average umbrella force cancels the average internal force acting on the primed coordinate, and thus is equal to the average derivative of the total potential V = V (r , r ) in the umbrella ensemble: dr dr [∂V /∂r i ] exp[−βV − βV u ] ∂V u F u = . (7.71) = ∂r i u dr dr exp[−βV − βV u ] In the nominator of (7.71) the term exp(βV u ) can be taken out of the integration over r and the remainder can be replaced by the derivative 11

K¨ astner and Thiel (2005) describe a method for the determination of the derivative of V mf .

7.7 Methods to derive the potential of mean force

237

of V mf (8.11):

We now obtain F u = u

∂V −βV ∂V mf e dr = ∂r i ∂r i

e−βV dr .

dr exp(−βV u )[∂V mf /∂r i ] dr exp(−βV ) , dr dr exp(−βV − βV u )

(7.72)

(7.73)

from which (7.70) follows. • By particle insertion by the method of Widom (1963) in the special case that the potential of mean force is a function of the position of a speciﬁc particle type and in addition the density of the medium is low enough to allow successful insertions. Let us consider the case that the potential of mean force of a system that is inhomogeneous in the z-direction but homogeneous in the x, y-plane (e.g., containing a planar phase boundary situated at z = 0 between two immiscible ﬂuids, or between a ﬂuid and a polymer, or containing a membrane between two liquid phases). Assume an equilibrium simulation is available. One may place a ghost particle (not exerting any forces on its environment) at a given z- but randomly chosen x, y-position in a conﬁguration of the generated ensemble, and registers the total interaction energy V between the particle and the surrounding molecules. This insertion is repeated many times, and μex (z) = −kT lnexp(−βV )z is determined. The standard chemical potential μ0 (z) of the particle type at position z in the inhomogeneous system is equal to the standard chemical potential μ0id of that particle type as an ideal gas, plus the measured excess μex (z); the potential of mean force as a function of z is, but for an additive constant, equal to μex (z). One may choose the number of insertions per value of z to satisfy statistical requirements. • By directly measuring the particle concentration c(z) (as a number density per unit volume) in an equilibrium simulation in the special case (as above) that the potential of mean force is a function of the position of a speciﬁc particle type, in regions where that concentration is high enough to allow its determination with suﬃcient accuracy. In those regions the potential of mean force can be constructed from V mf (z) = const − kT ln c(z).

(7.74)

• By enforcing the system to move from one part of conﬁgurational space to another. In such pulling simulations, also called steering molecular

238

Free energy, entropy and potential of mean force

dynamics (SMD),12 an extra external force is exerted on the system such that it will move much more quickly over intervening barriers than it would do in an equilibrium simulation. The advantage is that the steering can be executed in a global way without the need for a detailed description of the reaction coordinate; e.g., one may choose to distribute the pulling force over many atoms and let the system decide to ﬁnd the easiest pathway under such a global force. The disadvantage is that the work exerted by the external force contains a frictional contribution and is therefore not related to the potential of mean force in a straightforward way. Only in the uninteresting limit of zero pulling rate the force acts in a reversible manner and equals the derivative of a potential of mean force. However, in the case of small, but ﬁnite forces or pulling rates, it is possible to derive the potential of mean force V mf from the work W exerted by the external force. It is necessary to use a thermostat that prevents local heating by frictional forces. Although W will always exceed the reversible work ΔV mf , it can be shown (Jarzynsky, 1997a, 1997b) that e−βΔV

mf

= e−βW ,

(7.75)

which is valid if a suﬃciently large set of pulling simulations, starting from samples from an initial equilibrium ensemble, is performed. It is quite diﬃcult to obtain suﬃcient statistics for the evaluation of exp(−βW ); Park et al. (2003) found that more reliable results are obtained with a second-order expansion: lne−βW = −βW +

β2 2 W − W 2 , 2

(7.76)

using a time-dependent external potential of the form V ext =

k [ξ(r) − λ]2 , 2

λ(t) = λ0 + vt,

(7.77)

with a large force constant k. The method is very similar to the imposition of a slowly changing constraint. Except for near-equilibrium SMD, the method is not preferred above thermodynamic integration using constrained or umbrella-restrained intermediate simulations. In the next sec12

Simulations involving pulling were ﬁrst performed with the explicit purpose to mimic experiments with the atomic force microscope (AFM), e.g., pulling a ligand bound to a protein away from the protein. In such simulations the external force is due to a spring that is slowly displaced, like the lever of an AFM, but for the computation of a potential of mean force the spring is not required. The ﬁrst simulation of this kind was on the “unbinding” of biotin from the protein streptavidin by Grubm¨ uller et al. (1996), soon followed by Izrailev et al. (1997). See also Berendsen (1996). The name Steering Molecular Dynamics originates from Schulten, see Lu et al. (1998). The topic was reviewed by Isralewitz et al. (2001).

7.8 Free energy from non-equilibrium processes

239

tion the non-equilibrium methods are more fully treated and a proof of Jarzynski’s equation is given.

7.8 Free energy from non-equilibrium processes In the previous section we have seen how free energies or potentials of mean force can be computed through perturbation and integration techniques. The steered dynamics is reminiscent of the slow-growth methods, where an external agent changes the Hamiltonian of the system during a prolonged dynamics run, by adding an additional time-dependent potential or force, or changing the value of a constraint imposed on the system. Thus the system is literally forced from one state to another, possibly over otherwise unsurmountable barriers. If such changes are done very slowly, such that the system remains eﬀectively in equilibrium all the time, the change is a reversible process and in fact the change in free energy from the initial to the ﬁnal state is measured by the work done to change the Hamiltonian. In most practical cases the external change cannot be realized in a suﬃciently slow fashion, and a partial irreversible process results. The ensemble “lags behind” the change in the Hamiltonian, and the work done on the system by the external agent that changes the Hamiltonian, is partially irreversible and converted to heat. The second law of thermodynamics tells us that the total work W done on the system can only exceed the reversible part ΔA: W ≥ ΔA.

(7.78)

This is an inequality that enables us to bracket the free energy change between two measured values when the change is made both in the forward and the backward direction, but it does not give any help in quantifying the irreversible part. It would be desirable to have a quantitative relation between work and free energy! Such a relation indeed exists. Jarzynski (1997a, 1997b) has shown that for an irreversible process the Helmholtz free energy change follows from the work W done to change Hamiltonian H(λ) of the system from λ = 0 to λ = 1, if averaged over an equilibrium ensemble of initial points for λ = 0: A1 − A0 = −kB T lne−βW λ=0 .

(7.79)

This is the remarkable Jarzynski equation, which at ﬁrst sight is a counterintuitive expression, relating a thermodynamic quantity to a rather ill-deﬁned and very much process-dependent amount of work. Cohen and Mauzerall

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Free energy, entropy and potential of mean force

(2004) have criticized Jarzynski’s derivation on the basis of improper handling of the heat exchange with a heat bath, which induced Jarzynski (2004) to write a careful rebuttal. Still, the validity of this equation has been conﬁrmed by several others for various cases and processes, including stochastic system evolution (Crooks, 2000; Hummer and Szabo, 2001; Schurr and Fujimoto, 2003; Ath`enes, 2004). Since the variety of proofs in the literature is confusing, we shall give a diﬀerent proof below, which follows most closely the reasoning of Schurr and Fujimoto (2003). This proof will enable us to specify the requirements for the validity of Jarzynskyi’s equation. In this proof we shall pay extra attention to the role of the temperature, clarifying what requirements must be imposed on β.

7.8.1 Proof of Jarzynski’s equation Consider a system of interacting particles with Hamiltonian H0 that has been allowed to come to equilibrium with an environment at temperature T0 or Boltzmann parameter β0 = (kB T )−1 and has attained a canonical distribution exp[−β0 H0 (z)] . (7.80) p0 (z) = exp[−β0 H0 (z )] dz Here z stands for the coordinates (spatial coordinates and conjugate momenta) q1 , . . . , p1 , . . . of a point in phase space. At time t0 we pick a sample from the system with phase space coordinates z0 . When we speak later about averaging over the initial state, we mean averaging over the canonical distribution p0 of z0 . Now the system undergoes the following treatment (see Fig. 7.5): at time t0 the Hamiltonian is abruptly changed from H0 to H1 by an external agent; from t0 to t1 the system is allowed to evolve from z0 to z1 under the (constant) Hamiltonian H1 . The evolution is not necessarily a pure Hamiltonian evolution of the isolated system: the system may be coupled to a thermostat and/or barostat or extended with other variables, and the process may be deterministic or stochastic. The only requirement is that the evolution process conserves a canonical distribution exp[−β1 H1 (z)], where H1 (z) is the total energy of the system at phase point z. Note that we do not require that the temperature during evolution (e.g., given by the thermostat or the friction and noise in a stochastic evolution) equals the temperature before the jump. Now at t1 the external agent changes the Hamiltonian abruptly from H1 to H2 , after which the system is allowed to evolve under H2 from t1 to t2 , changing from z1 to z2 . Again, the evolution process is such that it

7.8 Free energy from non-equilibrium processes t0

Time Hamiltonian

H0

H1

Work Evolution in phase space

t1

G0 (β0 )

t2

H2

241 t3

-

H3

H4

-

-

-

-

W01

W12

W23

W34

z0

G1 (β1 )

-

G2 (β2 ) z1 z2

G3 (β3 )

z3

G4 (β4 )

Figure 7.5 Irreversible evolution with changing Hamiltonian. The Hamiltonian is abruptly changed by an external agent at times t0 , t1 , . . ., who exerts work W01 , W12 , . . . on the system. In the intervening time intervals the system is allowed to evolve with the propagator Gi (βi ), when the Hamiltonian Hi is valid. The points in phase space, visited by the system at t0 , t1 , . . ., comprising all coordinates and momenta, are indicated by z0 , z1 , . . ..

would conserve a canonical distribution exp[−β2 H2 (z)]. These processes of autonomous evolution followed by a Hamiltonian change may be repeated as often as required to reach the desired end state. Two deﬁnitions before we proceed: the work done by the external agent to change Hi to Hi+1 we deﬁne (following Jarzynski) as Wi,i+1 . But with changing β we also require the change in work relative to the temperature, which we shall denote by Fi,i+1 : Wi,i+1

def

=

Hi+1 (zi ) − Hi (zi ),

(7.81)

Fi,i+1

def

βi+1 Hi+1 (zi ) − βi Hi (zi ).

(7.82)

=

The sum of these quantities over more than one step is similarly denoted. For example, W0,i is the total work done in all steps up to and including the step to Hi , and F0,i is the total relative work done in all steps up to and including the step to Hi . These quantities have speciﬁed values for each realization; useful results require averaging over an initial distribution. In the following we shall prove that after each Hamiltonian jump to Hi , the free energy Ai is given by βi Ai − β0 A0 = − lne−F0,i p0 ,

(7.83)

Averaging is done over the initial canonical distribution p0 of z0 , and also

242

Free energy, entropy and potential of mean force

over all possible stochastic evolutions. The latter averaging is automatically fulﬁlled when the original distribution is suﬃciently sampled. This is the generalized Jarzynski equation. It reduces to the original Jarzynski equation when all βi are equal: Ai − A0 = −kB T lne−βW0,i p0 .

(7.84)

Hi can be taken as the required end state; the process may contain any number of steps and the intermediate evolution times may have any value from zero (no evolution) to inﬁnite (evolution to complete equilibrium). So the allowed processes encompass the already well-known single-step (no evolution) free energy perturbation, the few-step perturbation with intermediate equilibrations, and the limiting slow-growth process, the latter taking a large number of steps, each consisting of a small Hamiltonian change followed by a single MD time step. The proof follows by induction. Consider the free energy change after the ﬁrst step, before any evolution has taken place: dz0 exp[−β1 H1 (z0 )] β1 A1 − β0 A0 = − ln dz0 exp[−β0 H0 (z0 )] = e−[β1 H1 (z0 )−β0 H0 (z0 )] p0 = e−F0,1 p0 ,

(7.85)

which is the generalized Jarzynski’s equation applied to a single step without evolution. Now, from t0 to t1 the system is left to evolve under the Hamiltonian H1 . Its evolution can be described by a propagator G1 (z, t; z0 , t0 ; β1 ) that speciﬁes the probability distribution of phase points z at time t, given that the system is at z0 at time t0 . For pure Hamiltonian dynamics the path is deterministic and thus G1 is a delta-function in z; for stochastic evolutions G1 speciﬁes a probability distribution. In general G1 describes the evolution of a probability distribution in phase space: p(z, t) = G1 (z, t; z0 , t0 ; β1 )p(z0 , t0 ) dz0 . (7.86) The requirement that G1 preserves a canonical distribution exp[−β1 H1 (z)] can be written as G1 (z, t; z0 , t0 ) exp[−β1 H1 (z0 )] dz0 = exp[−β1 H1 (z)] (7.87) for all t. In fact, G maps the canonical distribution onto itself. The actual distribution p0 (z0 ) at t0 is not the canonical distribution for H1 , but rather the canonical distribution for H0 . So the property (7.87)

7.8 Free energy from non-equilibrium processes

z0(t0)

G1, H1

243

z1(t1)

Figure 7.6 Paths extending from sampled points z0 at t0 to z1 at t1 . Each of the paths is weighted (indicated by line thickness) such that the distribution of weights becomes proportional to the canonical distribution exp[−β1 H1 (z1 )]. The grey areas indicate the equilibrium distributions for H0 (left) and H1 (right).

cannot be applied to the actual distribution at t1 . But we can apply a trick, pictured schematically in Fig. 7.6. Let us give every point z0 a weight such that the distribution of weights, rather than of points, becomes the canonical distribution for H1 . This is accomplished if we give the point z0 a weight exp[−β1 H1 (z0 ) + β0 H0 (z0 )]. Note that this weight equals exp[−F01 ]. Since the distribution of weights, indicated by pw (z0 ), is now proportional to the canonical distribution for H1 : pw (z0 ) = p0 (z0 )e−β1 H1 (z0 )+β0 H0 (z0 ) =

exp[−β1 H1 (z0 )] , dz0 exp[−βH0 (z0 )]

(7.88)

the distribution of weights will remain invariant during the evolution with G1 to z1 , and hence also pw (z1 ) = pw (z0 ).

(7.89)

¿From this we can derive the unweighted distribution of points z1 by dividing pw (z1 ) with the weight given to z0 : p(z1 ) = pw (z1 )eβ1 H1 (z0 )−β0 H0 (z0 )] = pw (z1 )eF01 exp[−β1 H1 (z1 ) + F01 ] . = dz0 exp[−βH0 (z0 )]

(7.90)

Next the external agent changes H1 to H2 , performing the work W1,2 = H2 (z1 ) − H1 (z1 ) on the system. The relative work is F1,2 = β2 H2 (z1 ) − β1 H1 (z1 ).

(7.91)

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Free energy, entropy and potential of mean force

Equation (7.90) can now be rewritten as p(z1 ) =

exp[−β2 H2 (z1 ) + F0,1 + F1,2 ] . dz0 exp[−βH0 (z0 )]

If we now ask what the expectation of exp[−F0,2 ] will be, we ﬁnd e−F0,2 = e−(F0,1 +F1,2 ) = dz1 p(z1 )e−(F0,1 +F1,2 ) dz1 exp[−β2 H2 (z1 )] = e−(β2 A2 −β1 A0 ) . = dz0 exp[−βH0 (z0 )]

(7.92)

(7.93) (7.94)

This is the generalized Jarzynski’s equation after the second step has been made. The extension with subsequent steps is straightforward: for the next step we start with p(z1 ) and give the points a weight exp[−F0,2 ]. The weight distribution is now the canonical distribution for H2 , which remains invariant during the evolution G2 . From this we derive p(z2 ), and – after having changed the Hamiltonian at t2 to H3 – we ﬁnd that e−F0,3 = e−(β3 A3 −β0 A0 ) .

(7.95)

This, by induction, completes the proof of (7.83). Note that, in the case of varying β during the process, it is the total relative work, i.e., the change in energy divided by the temperature, F0,i , that must be exponentially averaged rather than the total work itself.

7.8.2 Evolution in space only When the external change in Hamiltonian involves the potential energy V (r) only (which usually is the case), and the evolution processes are mappings in conﬁgurational space that conserve a canonical distribution (e.g., a sequence of Monte Carlo moves or a Brownian dynamics), the Jarzynski equation is still valid. The evolution operator Gi (r i , ti ; r i−1 , ti−1 ; βi ) now evolves r i−1 into r i ; it has the property (7.96) Gi (r , t ; r, t; βi ) exp[−βi Vi (r)] dr = exp[−β1 Vi (r )]. Here r stands for all cartesian coordinates of all particles, specifying a point in conﬁgurational space. When we re-iterate the proof given above, replacing z by r and H by V , we ﬁnd the same equation (7.84) for the isothermal case, but a correction due to the kinetic contribution to the free energy if

7.8 Free energy from non-equilibrium processes

245

the initial and ﬁnal temperatures diﬀer. Equation (7.83) is now replaced by βi Ai − β0 A0 =

βi 3N ln − lne−F0,i p0 . 2 β0

(7.97)

7.8.3 Requirements for validity of Jarzynski’s equation Reviewing the proof given above, we can list the requirements for its validity: (i) The state of the system at time t is completely determined by the point in phase space z(t). The propagator G determines the future probability distribution, given z(t). This is an expression of the Markovian character of the propagator: the future depends on the state at t and not on earlier history. This precludes the use of stochastic propagators with memory, such as the generalized Langevin equation. It is likely (but not further worked out here) that the Markovian property is not a stringent requirement, as one can always deﬁne the state at time t to include not only z(t), but also z at previous times. However, this would couple the Hamiltonian step with the future propagation, with as yet unforeseen consequences. Hamiltonian (including extended systems), simple Langevin, Brownian and Monte Carlo propagations are all Markovian.13 (ii) The propagator must have the property to conserve a canonical distribution. Microscopic reversibility and detailed balance are not primary requirements. (iii) The sampling must be suﬃcient to eﬀectively reconstruct the canonical distribution after each step by the weighting procedure. This requires suﬃcient overlap between the distribution of end points of each relaxation period and the canonical distribution after the following step in the Hamiltonian. When the steps are large and the relaxations are short, suﬃcient statistics may not be available. This point is further discussed in the next subsection. (iv) As is evident from the proof, there is no requirement to keep the inverse temperature β constant during the process. Even if the same β is required for the initial and ﬁnal states, intermediate values may be chosen diﬀerently. This property may be exploited to produce a faster sampling. 13

See Park and Schulten (2004) for a discussion of various ensembles.

246

Free energy, entropy and potential of mean force

7.8.4 Statistical considerations Since the averaging is done over an exponential function of the work done, the trajectories with the smaller work values will dominate the result. This produces erratic jumps in cumulative averages whenever occasional low values appear. The statistical properties and validity of approximations have been considered by several authors (Park et al. 2003, Hummer 2001, Ytreberg and Zuckerman 2004). Let us consider a simple example. We sample a property x (the “work”) from a distribution function p(x), and we wish to compute the quantity A: 1 1 A = − lne−βx = − ln p(x)e−βx dx. (7.98) β β Without loss of generality, we make take the average of x as zero, so that all values refer to the average of the distribution. First consider the cumulant expansion14 in powers of β, obtained from a simple Taylor expansion: 1 1 1 A = − βx2 + β 2 x3 − β 3 [x4 − 3x2 2 + · · ·]. 2! 3! 4!

(7.99)

For a Gaussian distribution, √

−1

p(x) = (σ 2π)

x2 exp − 2 , 2σ

(7.100)

only the ﬁrst term survives, as can easily be seen by direct integration: 1 A = − βσ 2 . 2

(7.101)

Figure 7.7 shows that the cumulative average gives very poor convergence. For this ﬁgure 1000 points have been sampled from a normal distribution of zero average and unit variance, and the cumulative exponential averages were calculated with values of β equal to 1, 2 and 4. The theoretical values for A are −0.5, −1 and −2, respectively. For β = 1 convergence is reached after about 600 points; for β = 2 1000 points are barely enough, and for β = 4 1000 points are clearly insuﬃcient to reach convergence. This means that computing the exponential average is hardly an option if the computation of one path takes a considerable computational eﬀort. The route via the cumulant expansion (7.99) gives very accurate results if the distribution is known to be Gaussian and (7.101) applies. For n independent samples, the variance (mean-square error) in the estimated average 14

See Zwanzig (1954), who deﬁned the cumulant expansion of lnexp(−βx) in a power series in (−β)n /n!.

7.8 Free energy from non-equilibrium processes

247

A

0 β=1 β=2

−1

β=4 −2

−3

200

400

600

800

1000 n

Figure 7.7 Cumulative average of A = −β −1 lnexp(−βx) over n samples drawn from a normal distribution (average 0, variance σ 2 = 1). The theoretical limits are −0.5β, indicated by dotted lines.

x = xi /n is σ 2 /n, while the mean-square error in the estimated variance s2 = (x − x)2 /(n − 1) is 2σ 4 /(n − 1) (Hummer, 2001): 1/2 2 β2σ 1 2 σ ˆ ˆ4 ˆ ± + A = x − β σ , (7.102) 2 n n−1 where x = σ ˆ2 =

1 xi n 1 n−1

(7.103) (xi − x)2 ,

(7.104)

are best, unbiased, estimates for the average and variance. However, if the distribution is not Gaussian, the higher-order cumulants rapidly add to the inaccuracy. Ytreberg and Zuckerman (2004) propose, based on an extensive error analysis, to select many random sequences of m < n samples from a set of n data and plot the exponential averages thus obtained versus m−1/2 . Extrapolation to n → ∞ then corrects for a datasize-dependent systematic bias in the averages, and an error estimate can be obtained.

248

Free energy, entropy and potential of mean force

In practice, Jarzynski’s equation can only be used if the irreversible work is small (not exceeding 2kB T ), i.e., if the process is close to equilibrium. As Oostenbrink and van Gunsteren (2006) have shown, integration from equilibrated intermediate points is generally much more eﬃcient than both irreversible fast growth and near-equilibrium slow growth. It is not clear whether this still holds when optimal corrections are applied to the integration by fast or slow growth.

8 Stochastic dynamics: reducing degrees of freedom

8.1 Distinguishing relevant degrees of freedom Often the interest in the behavior of large molecular systems concerns global behavior on longer time scales rather than the short-time details of local dynamics. Unfortunately, the interesting time scales and system sizes are often (far) beyond what is attainable by detailed molecular dynamics simulations. In particular, macromolecular structural relaxation (crystallization from the melt, conformational changes, polyelectrolyte condensation, protein folding, microphase separation) easily extends into the seconds range and longer. It would be desirable to simplify dynamical simulations in such a way that the “interesting” behavior is well reproduced, and in a much more eﬃcient manner, even if this goes at the expense of “uninteresting” details. Thus we would like to reduce the number of degrees of freedom that are explicitly treated in the dynamics, but in such a way that the accuracy of global and long-time behavior is retained as much as possible. All approaches of this type fall under the heading of coarse graining, although this term is often used in a more speciﬁc sense for models that average over local d etails. The relevant degrees of freedom may then either be the cartesian coordinates of special particles that represent a spatial average (the superatom approach, treated in Section 8.4), or they may be densities on a grid, deﬁned with a certain spatial resolution. The latter type of coarse graining is treated in Chapter 9 and leads to mesoscopic continuum dynamics, treated in Chapter 10. The ﬁrst choice is to distinguish relevant degrees of freedom from irrelevant degrees of freedom. With “irrelevant” we do not mean unimportant: these degrees of freedom can have essential inﬂuences on the “relevant” degrees of freedom, but we mean that we don’t require knowledge of the detailed behavior of those degrees of freedom. This choice is in a sense arbitrary and 249

250

Stochastic dynamics: reducing degrees of freedom

depends on the system, the properties of interest and the required accuracy. The choice must be judiciously made. It is highly desirable and beneﬁcial for the approximations that will be made, that the “irrelevant” degrees of freedom equilibrate faster (and preferably much faster) than the “relevant” degrees of freedom, as the approximations will unavoidably produce errors on the time scale were these two overlap. However, such a clear distinction is generally not possible, and one must accept the inaccurate prediction of dynamic details of the “relevant” degrees of freedom on short time scales. Some examples are listed below: • A rather spherical molecule, as CCl4 . Relevant: the center-of-mass motion; irrelevant: the rotational and internal degrees of freedom. • A (macro)molecule in a solvent. Relevant: all atoms of the solute; irrelevant: all solvent molecules. With this choice there is certainly overlap between the time ranges for the two sets of particles. For example, for a protein in water one may expect incorrect dynamic behavior of charged side chains on a time scale shorter than, or comparable to, the dielectric relaxation time of the solvent. • A large linear polymer. Relevant: the centers of mass of groups of n consecutive atoms; irrelevant: all other degrees of freedom. This may work if the polymer shows self-similar behavior, i.e., that macroscopic properties scale in some regular manner with n. These are typical superatom models (see Section 8.4). • A protein (or other compact non-selfsimilar macromolecule). Relevant: a subset of atoms (as Cα atoms, or backbone atoms, or backbone atoms plus a simpliﬁed side chain representation); irrelevant: all other atoms or degrees of freedom including the surrounding solvent. This is also a superatom approach. • A protein (or other compact macromolecule). Relevant: a set of collective “essential degrees of freedom” generated from an analysis of a detailed simulation, e.g., the ﬁrst few eigenvectors with largest eigenvalue from a principal component analysis based on atomic ﬂuctuations, or from a quasi-harmonic analysis. Irrelevant: all other eigenvectors. • A chemical reaction or other infrequent process in a complex system. Relevant: the reaction coordinate, being a function of internal degrees of freedom of the system that captures the important path between reactants and products in a chemical reaction. This may concern one dimension, or encompass a space of a few dimensions. Irrelevant: all other degrees of freedom. • A colloidal dispersion of relatively large spherical rigid particles in a sol-

8.2 The generalized Langevin equation

251

vent. Relevant: center of mass coordinates of the particles. Irrelevant: rotational and internal degrees of freedom of the particles, and solvent degrees of freedom. For non-spherical particles their rotational degrees of freedom may be considered relevant as well. • A rather homogeneous condensed phase under slowly varying external inﬂuences. Relevant: Densities of molecular components at grid points on a chosen 3D spatial grid; irrelevant: all other degrees of freedom. Instead of a regular 3D grid one may use other, possibly time-dependent, ﬁnite element subdivisions of space. In some cases we can choose cartesian degrees of freedom as the relevant ones (e.g., when we divide the particles over both classes), but in most cases we must deﬁne the two classes as generalized degrees of freedom. To avoid unnecessary accumulation of complexity, we shall in the following consider cartesian coordinates ﬁrst, and consider necessary modiﬁcations resulting from the use of generalized coordinates later (Section 8.6.1 on page 263). In the case that the relevant coordinate is a distance between atoms or a linear combination of atomic coordinates, the equations are the same as for cartesian coordinates of selected particles, although with a diﬀerent eﬀective mass.

8.2 The generalized Langevin equation Assume we have split our system into explicit “relevant particles” indicated with a prime and with positions r i (t) and velocities v i (t), i = 1, . . . , N , and implicit double-primed “irrelevant” particles with positions r j (t) and velocities v j (t), j = 1, . . . , N . The force F i acting on the i-th primed particle comes partly from interactions with other primed particles, and partly from interactions with double-primed particles. The latter are not available in detail. The total force can be split up into: • systematic forces F si (r ) which are a function of the primed coordinates; these forces include the mutual interactions with primed particles and the interactions with double-primed particles as far as these are related to the primed positions; • frictional forces F fi (v) which are a function of the primed velocities (and may parametrically depend on the primed coordinates as well). They include the interactions with double-primed particles as far as these are related to the primed velocities; • random forces F ri (t). These are a representation of the remainder of the interactions with double-primed particles which are then neither related to

252

Stochastic dynamics: reducing degrees of freedom

the primed positions nor to the primed coordinates. Such forces are characterized by their statistical distributions and by their time correlation functions. They may parametrically depend on the primed coordinates. This classiﬁcation is more intuitive than exact, but suﬃces (with additional criteria) to derive these forces in practice. A systematic way to derive the time evolution of a selected subsystem in phase space is given by the projection operator formalism of Zwanzig (1960, 1961, 1965) and Mori (1965a, 1965b). This formalism uses projection operators in phase space, acting on the Liouville operator, and arrives at essentially the same subdivision of forces as given above. In this chapter we shall not make speciﬁc use of it. The problem is that the formalism, although elegant and general, does not make it any easier to solve practical problems.1 We make two further assumptions: (i) the systematic force can be written as the gradient of a potential in the primed coordinate space. This is equivalent to the assumption that the systematic force has no curl. For reasons that will become clear, this potential is called the potential of mean force, V mf (r ); (ii) the frictional forces depend linearly on velocities of the primed particles at earlier times. Linearity means that velocity-dependent forces are truncated to ﬁrst-order terms in the velocities, and dependence on earlier (and not future) times is simply a result of causality. Now we can write the equations of motion for the primed particles as ∂V mf t dv i mi =− − ζij (τ )v j (t − τ ) dτ + η i (t), (8.1) dt ∂r i 0 j

where ζij (τ ) (often written as mi γij (τ )) is a friction kernel that is only deﬁned for τ ≥ 0 and decays to zero within a ﬁnite time. The integral over past velocities extends to τ = t, as the available history extends back to time 0; when t is much larger than the correlation time of ζij (τ ), the integral can be safely taken from 0 to ∞. This friction term can be viewed as a linear prediction of the velocity derivative based on knowledge of the past trajectory. The last term η(t) is a random force with properties still to be determined, but surely with η(t) = 0,

(8.2)

v i (t) · η j (t ) = 0 for any i, j and t ≥ t. 1

(8.3)

van Kampen (1981, pg 398) about the resulting projection operator equation: “This equation is exact but misses the point. [. . . ] The distribution cannot be determined without solving the original equation . . . ”

8.2 The generalized Langevin equation

253

On the time scale of the system evolution, the random forces are stationary stochastic processes, independent of the system history, i.e., their correlation functions do not depend on the time origin, although they may have a weak dependence on system parameters. In principle, the random forces are correlated in time with each other; these correlations are characterized by η correlation functions Cij (τ ) = η i (t)η j (t + τ ), which appear (see below) to be related to the friction kernels ζij (τ ). This is the generalized Langevin equation for cartesian coordinates. For generalized coordinates {q, p} the mass becomes a tensor; see Section 8.6.1 on page 263. Note At this point we should make two formal remarks on stochastic equations (like (8.1)) of the form dy = f (y) + cη(t), (8.4) dt where η(t) is a random function. The ﬁrst remark concerns the lack of mathematical rigor in this equation. If the random function represents white noise, it can be seen as a sequence of delta functions with random amplitudes. Every delta function causes a jump in y and the resulting function y(t) is not diﬀerentiable, so that the notation dy/dt is mathematically incorrect. Instead of a diﬀerential equation (8.4), we should write a diﬀerence equation for small increments dt, dy, dw: dy = f (y) dt + c dw(t),

(8.5)

where w(t) is the Wiener–L´evy process, which is in fact the integral of a white noise. The Wiener-L´evy process (often simply called the Wiener process is nonstationary, but its increments dw are stationary normal processes. See, for example, Papoulis (1965) for deﬁnitions and properties of random processes. While modern mathematical texts avoid the diﬀerential equation notation,2 this notation has been happily used in the literature, and we shall use it as well without being disturbed by the mathematical incorrectness. What we mean by a stochastic diﬀerential equation as (8.4) is that the increment of y over a time interval can be obtained by integrating the right-hand side over that interval. The integral r of the random process η(t) t+Δt r= η(t ) dt (8.6) t

is a random number with zero mean and variance given by a double integral over the correlation function of η: t+Δt t+Δt dt dt η(t )η(t ); (8.7) r2 = t

t

in the case of a white noise η(t )η(t ) = δ(t − t ) and therefore r2 = Δt. The other remark concerns a subtlety of stochastic diﬀerential equations with a y-dependent coeﬃcient c(y) in front of the stochastic white-noise term: the widely 2

See, e.g., Gardiner (1990). An early discussion of the inappropriateness of stochastic diﬀerential equations has been given by Doob (1942).

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Stochastic dynamics: reducing degrees of freedom

debated Itˆ o–Stratonovich “dilemma.”3 Solving the equation in time steps, the variable y will make a jump every step and it is not clear whether the coeﬃcient c(y) should be evaluated before or after the time step. The equation therefore has no meaning unless a recipe is given how to handle this dilemma. Itˆ o’s recipe is to evaluate c(y) before the step; Stratonovich’s recipe is to take the average of the evaluations before and after the step. The stochastic equation is meant to deﬁne a process that will satisfy a desired equation for the distribution function P (y, t) of y. If that equation reads ∂P ∂ 1 ∂2 = − f (y)P + c(y)2 P, ∂t ∂y 2 ∂y 2

(8.8)

Itˆo’s interpretation appears to be correct. With Stratonovich’s interpretation the last term is replaced by 1 ∂ ∂ c(y) c(y)P. 2 ∂y ∂y

(8.9)

Hence the derivation of the equation will also provide the correct interpretation. Without that interpretation the equation is meaningless. As van Kampen (1981, p. 245) remarks: “no amount of physical acumen suﬃces to justify a meaningless string of symbols.” However, the whole “dilemma” arises only when the noise term is white (i.e., when its time correlation function is a delta function), which is a mathematical construct that never arises in a real physical situation. When the noise has a non-zero correlation time, there is no diﬀerence between Itˆo’s and Stratonovich’s interpretation for time steps small with respect to the correlation time. So, physically, the “dilemma” is a non-issue after all.

In the following sections we shall ﬁrst investigate what is required for the potential of mean force in simulations that are meant to preserve long-time accuracy. Then we describe how friction and noise relate to each other and to the stochastic properties of the velocities, both in the full Langevin equation and in the simpler pure Langevin equation which does not contain the systematic force. This is followed by the introduction of various approximations. These approximations involve both temporal and spatial correlations in the friction and noise: time correlations can be reduced to instantaneous response involving white noise and spatial correlations can be reduced to local terms, yielding the simple Langevin equation. In Section 8.7 we average the Langevin dynamics over times long enough to make the inertial term negligible, yielding what we shall call Brownian dynamics.4 3 4

See van Kampen (1981) and the original references quoted therein. The term Brownian dynamics in this book is restricted to approximations of particle dynamics that are inertia-free but still contain stochastic forces. There is no universal agreement on this nomenclature; the term Brownian dynamics is sometimes used for any dynamical method that contains stochastic terms.

8.3 The potential of mean force

255

8.3 The potential of mean force In a system with reduced dimensionality it is impossible to faithfully retain both thermodynamic and dynamic properties on all time and length scales. Since the interest is in retaining properties on long time scales and with coarse space resolution, we shall in almost all cases be primarily interested in a faithful representation of the thermodynamic properties at equilibrium, and secondarily in the faithful representation of coarse-grained non-equilibrium behavior. If we can maintain these objectives we should be prepared to give up on accurate local and short-time dynamical details. The criterium of retaining thermodynamic accuracy prescribes that the partition function generated by the reduced dynamics should at least be proportional to the partition function that would have been generated if all degrees of freedom had been considered. Assuming canonical ensembles, this implies that the probability distribution w(r ) in the primed conﬁgurational space should be proportional to the integral of the Boltzmann factor over the double-primed space: (8.10) w(r ) ∝ e−βV (r ,r ) dr . Now we deﬁne the potential of mean force as V mf (r ) = −kT ln e−βV (r ,r ) dr ,

(8.11)

which implies that w(r ) dr ∝ e−βV

mf (

r ) dr .

(8.12)

It follows by diﬀerentiation that the forces derived as a gradient of V mf equal the exact force averaged over the ensemble of primed coordinates: (∂V (r , r )/∂r ) exp[−βV (r , r )] dr mf ∇r V = . (8.13) exp[−βV (r , r ) dr ] Note that V mf also contains the direct interactions between r ’s, which are separable from the integral in (8.11). It is a free energy with respect to the double-primed variables (beware that therefore V mf is temperaturedependent!), but it still is a function of the primed coordinates. It determines in a straightforward manner the probability distribution in the primed space. Note that V mf is not a mean potential over an equilibrium double-primed ensemble: V mf = V (r , r ) .

(8.14)

Whereas several methods are available to compute potentials of mean force

256

Stochastic dynamics: reducing degrees of freedom

from simulations, as is treated in detail in Chapter 7, Section 7.5 on page 227, empirical validation and generally adjustments are always necessary; the best results are often obtained with completely empirical parametrization because the model can then be ﬁne-tuned to deliver the thermodynamic accuracy required for the application. In the next section we consider the special case of superatom models.

8.4 Superatom approach A special form of coarse graining is the representation of local groups of atoms by one particle, called a superatom. Superatoms are especially useful in chain molecules as polymers and lipids, where they typically represent three to ﬁve monomer units. This is a compromise between accuracy and simulation eﬃciency. Several superatom deﬁnitions have been published and most applications do not include additional friction and noise to represent the forces due to the left-out degrees of freedom. This should not inﬂuence the equilibrium properties, but is likely to yield a faster dynamics than the real system. The “bead-and-spring” models for polymers, which have a long history in polymer physics, are in fact also superatom models, although they were intended as prototype polymer models rather than as simpliﬁed representation of a speciﬁc real polymer.5 The interaction between neighboring beads in a chain is often simply a soft harmonic potential that leads to a Gaussian distribution for the distance between beads; since polymers chains can only be extended to an upper limit rm , a somewhat more realistic model is the FENE (ﬁnitely extendable nonlinear elastic) chain model, with a force between neighboring beads with interbead vector r given by r . (8.15) F = −H 1 − (r/rm )2 See Fan et al. (2003) for a stochastic application of the FENE model. More recently superatom models have been designed to speciﬁcally represent real molecules, e.g., alkanes by the coarse-grained model of Nielsen et al. (2003), with Lennard–Jones (LJ) superatoms representing three nonhydrogen atoms. In addition to LJ, this model has soft harmonic bond length and bond angle potentials. The model is parameterized on density and surface tension of liquid alkanes and reproduces end-to-end distribution functions obtained from simulations with atomic details. A more general and very successful coarse-grained force ﬁeld for lipids and surfactant systems has been deﬁned and tested by Marrink et al. (2004). It consists of four types of 5

See M¨ uller-Plathe (2002) for a review on multiscale modelling methods for polymers.

8.5 The ﬂuctuation–dissipation theorem

257

particles (charged, polar, non-polar, apolar) with the charged and non-polar types subdivided into four subtypes depending on their hydrogen-bonding capability. Each particle represents about four non-hydrogen atoms; also four water molecules are represented by one particle. The interactions are of Lennard–Jones and Coulomb type, smoothly switched-oﬀ at 1.2 nm, with ﬁve possible values for the ε and only one (0.47 nm) for the σ parameters of the LJ interaction. In addition there are fairly soft harmonic bond and bondangle interaction terms; the total number of parameters is eight. Despite its simplicity, the model reproduces density and isothermal compressibility of water and liquid alkanes within 5% and reproduces mutual solubilities of alkanes in water and water in alkanes to within a free energy of 0.5kB T . The melting point of water is 290 K. A time step of 50 fs is possible, and since the dynamics of this model (no friction and noise are added) is about four times faster than reality, an eﬀective time step of 0.2 ps can be realized. One then easily simulates real systems with millions of atoms over microseconds, allowing the study of lipid bilayer formation, micellar formation, vesicle formation and gel/liquid crystalline phase changes with realistic results.6 See Fig. 8.1 for comparison of a coarse-grained and a detailed simulation of the spontaneous formation of a small lipid vesicle.

8.5 The ﬂuctuation–dissipation theorem

Let us look at the long-time behavior of the kinetic energy K = i 21 mi vi2 in the generalized Langevin equation (8.1). The ﬁrst term on the r.h.s. (the systematic force) simply exchanges kinetic and potential energy, keeping the total energy constant. The second term (friction or dissipation) reduces the kinetic energy and the third stochastic term (noise) increases the kinetic energy. In order for the process to be stationary, the velocity correlation functions v i (t)v j (t + τ ) should become independent of t for large t; in particular the average squared velocities v i (t)v j (t), which are thermodynamic quantities, should fulﬁll the equipartition theorem viα (t)vjβ (t) =

kB T δij δαβ . mi

(8.16)

That is, if the random process is realized many times from the same starting conﬁguration at t = 0, then after a suﬃciently long time – when the memory of the initial conditions has decayed – the average over all realizations should 6

Spontaneous aggregation of lipid bilayers, gel/liquid crystalline transitions, inverted hexagonal phase formation and formation of Micelles: Marrink et al. (2004); hexagonal phase formation: Marrink and Mark (2004); vesicle fusion: Marrink and Mark (2003a); vesicle formation: Marrink and Mark (2003b).

258

Stochastic dynamics: reducing degrees of freedom

Figure 8.1 Two simulations of the spontaneous formation of a lipid bilayer vesicle. Upper panel: atomic detail molecular dynamics; lower panel: coarse-grained superatom dynamics simulation (Courtesy of S.-J. Marrink and A. H. de Vries, University of Groningen; reproduced by permission from J. Comput. Chem. (van der Spoel et al., 2005)

fulﬁll the equipartition theorem. This is one – rather restricted – formulation of the ﬂuctuation–dissipation theorem. The general ﬂuctuation–dissipation theorem relates the linear response of some system variable v to the spontaneous ﬂuctuation of v. Kubo (1966) distinguishes a ﬁrst and a second ﬂuctuation–dissipation theorem: the ﬁrst theorem says that the normalized time response of v to a δ-disturbance equals the normalized correlation function of the spontaneously ﬂuctuating v in equilibrium (see Section 18.3 on page 511); the second theorem relates the friction kernel ζ(t) to the correlation function of the random term η(t) in the Langevin equation.. To illustrate these theorems we apply for the sake of simplicity the response to a single velocity v(t) that is assumed to follow the pure Langevin equation without systematic force: t mv(t) ˙ =− ζ(τ )v(t − τ ) dτ + η(t) + F ext (t), (8.17) 0

F ext (t)

an external force meant to provide a disturbance to measure with the linear response. Now apply an external δ-force at time 0: F ext (t) = mv0 δ(t).

(8.18)

This produces a jump v0 in the velocity at time t = 0. The velocity v(t) subsequently evolves according to (8.17) (with the external force no longer

8.5 The ﬂuctuation–dissipation theorem

259

being present). What we call the δ-response v0 Φ(t) of the velocity7 (see (18.1) on page 507) is the ensemble average over many realizations of this process, with initial conditions taken randomly from an unperturbed equilibrium distribution and with independent realizations of the noise force η(t): v0 Φ(t) = v(t).

(8.19)

The ﬁrst ﬂuctuation–dissipation theorem states that Φ(t) =

v(t0 )v(t0 + t) , v 2

(8.20)

where the average is now taken over an unperturbed equilibrium ensemble, for which the velocity correlation function is stationary and hence independent of the time origin t0 . Proof From (8.19) it follows that v(t) ˙ = v0

dΦ , dt

(8.21)

so that on averaging (8.17) immediately gives an equation for Φ, considering that the average over the noise is zero (see (8.2)): t dΦ =− ζ(τ )Φ(t − τ ) dτ. (8.22) m dt 0 Given the friction kernel ζ(τ ) and the initial value Φ(0) = 1, this equation determines the response function Φ(t). def

The velocity autocorrelation function C v (t) = v(t0 )v(t0 + t) can be found by applying (8.17) to the time t0 + t, multiplying both sides with v(t0 ) and taking the ensemble average: t ˙ 0 +t) = − ζ(τ )v(t0 )v(t0 +t−τ ) dτ +v(t0 )η(t0 +t), (8.23) mv(t0 )v(t 0

which can be written in terms of the velocity correlation function C v (t), realizing that the last term vanishes because the random force does not correlate with velocities at earlier times (see (8.3)), as t d v ζ(τ )C v (t − τ ) dτ. (8.24) m C (t) = − dt 0 7

Including v0 in the response means that Φ is normalized: Φ(0) = 1.

260

Stochastic dynamics: reducing degrees of freedom

Given the friction kernel ζ(τ ) and the initial value C v (0) = v 2 , this equation determines the equilibrium correlation function C v (t). But this equation is equivalent to the corresponding equation (8.22) for Φ(t), from which it follows that C v (t) = v 2 Φ(t).

(8.25)

The ﬁrst ﬂuctuation–dissipation theorem has a solid basis; it applies in general to small deviations from equilibrium, also for systems that include systematic forces (see Section 18.3 on page 511). It is the basis for the derivation of transport properties from equilibrium ﬂuctuations. However, it does not provide the link between friction and noise needed for the implementation of Langevin dynamics. The second ﬂuctuation–dissipation theorem states that η(t0 )η(t0 + t) = mv 2 ζ(t) = kB T ζ(t).

(8.26)

This theorem provides the proper connection between friction and noise, but it stands on much weaker grounds than the ﬁrst theorem. It can be rigorously proven for a pure Langevin equation without systematic force. The proof uses Laplace transforms or one-sided Fourier transforms and rests on the derivation of the stationary velocity autocorrelation function, given the noise correlation function, which must equal the solution of (8.24). We refer the reader for this proof to the literature, where it can be found in several places; a readable discussion is given in the ﬁrst chapter of Kubo et al. (1985). When systematic non-linear forces are present (as is the case in all simulations of real systems), the theorem can no longer be proven to be generally valid. Various special cases involving harmonic forces and heat baths consisting of collections of harmonic oscillators have been considered,8 and modiﬁcations for the general case have been proposed.9 While the matter appears not to be satisfactorily settled, our recommendation is that time-dependent friction kernels should not be used in cases when intrinsic relaxation times, determined by the systematic forces, are of the same order as the characteristic times of the friction kernels. 8

9

The harmonic-oscillator heat bath was pioneered by Zwanzig (1973) and extended by Cohen (2002); Hernandez (1999) considered the projection of non-equilibrium Hamiltonian systems. Adelman and Doll (1974) simpliﬁed Zwanzig’s approach for application to atomic collisions with a solid surface. Ciccotti and Ryckaert (1981) separate the systematic force and obtain a modiﬁed friction and noise; Bossis et al. (1982) show that the eﬀect of the systematic force is a modiﬁcation of the second ﬂuctuation–dissipation theorem by the addition of an extra term equal to the correlation of velocity and systematic force. McDowell (2000) considers a chain of heat baths and concludes that an extra bias term should be added to the random force.

8.5 The ﬂuctuation–dissipation theorem

261

However, the memory-free combination of time-independent friction and white noise does yield consistent dynamics, also in the presence of systematic forces, with proper equilibrium ﬂuctuations. This memoryless approximation is called a Markovian process,10 and we shall call the corresponding equation (which may still be multidimensional) the Markovian Langevin equation. Consider, ﬁrst for the simple one-dimensional case, the change in kinetic energy due to a Markovian friction force and a white-noise stochastic force. The equation of motion is mv˙ = F (t) − ζv + η(t),

(8.27)

η(t0 )η(t0 + t) = Aη δ(t),

(8.28)

with

where Aη is the intensity of the noise force. Consider the kinetic energy K(t) = 12 mv 2 . The friction force causes a decrease of K: dK 2ζ (8.29) = mv v˙ = −ζv 2 = − K. dt friction m The stochastic term causes an increase in K. Consider a small time step Δt, which causes a change in velocity: t+Δt mΔv = η(t ) dt . (8.30) t

The change in K is 1 1 (ΔK)noise = m[(v + Δv)2 − v 2 ] = mvΔv + m(Δv)2 . (8.31) 2 2 We are interested in the average over the realizations of the stochastic process. The ﬁrst term on the r.h.s. vanishes as Δv is not correlated with v. The second term yields a double integral: t+Δt t+Δt 1 1 ΔKnoise = Aη Δt; dt dt η(t )η(t ) = (8.32) 2m t 2m t therefore the noise causes on average an increase of K: Aη dK . = dt noise 2m

(8.33)

Both of these changes are independent of the systematic force. They balance 10

A Markov process is a discrete stochastic process with transition probabilities between successive states that depend only on the properties of the last state, and not of those of previous states.

262

Stochastic dynamics: reducing degrees of freedom

on average when K = Aη /(4ζ). Using the equilibrium value at a reference temperature T0 for one degree of freedom: 1 K = kB T0 , 2

(8.34)

it follows that a stationary equilibrium kinetic energy is obtained for Aη = 2ζkB T0 .

(8.35)

If the initial system temperature T deviates from T0 = A/(2ζkB ), it will decay exponentially to the reference temperature T0 set by the noise, with a time constant m/2ζ: 2ζ dT = − (T − T0 ). (8.36) dt m Thus the added friction and noise stabilize the variance of the velocity ﬂuctuation and contribute to the robustness of the simulation. The ﬂow of kinetic energy into or out of the system due to noise and friction can be considered as heat exchange with a bath at the reference temperature. This exchange is independent of the systematic force and does not depend on the time dependence of the velocity autocorrelation function. It is easy to see that the latter is not the case for a time-dependent (non-Markovian) friction: (8.29) then reads ∞ dK = mv(t)v(t) ˙ =− ζ(τ )v(t)v(t − τ ) dτ, (8.37) dt friction 0 which clearly depends on the velocity correlation function, which in turn depends on the behavior of the systematic force. The whole purpose of simplifying detailed dynamics by Langevin dynamics is reducing fast degrees of freedom to a combination of friction and noise. When these “irrelevant” degrees of freedom indeed relax fast with respect to the motion of the relevant degrees of freedom, they stay near equilibrium under constrained values of the “relevant” degrees of freedom and in fact realize a good approximation of the constrained canonical ensemble that is assumed in the derivation of the systematic force (8.13) and that allows the simpliﬁcation without loosing thermodynamic accuracy. “Fast” means that the correlation time of the force due to the “irrelevant” degrees of freedom (the frictional and random force) is short with respect to the relaxation time within the “relevant” system, due to the action of the systematic force. The latter is characterized by the velocity correlation function in the absence of friction and noise. If the forces from the “irrelevant” degrees of freedom are fast in this sense, a Markovian friction and noise friction will be a good

8.6 Langevin dynamics

263

approximation that even preserves the slow dynamics of the system; if they are not fast, a Markovian Langevin simulation will perturb the dynamics, but still preserve the thermodynamics.

8.6 Langevin dynamics In this section we start with the generalized Langevin equation (8.1), which we ﬁrst formulate in general coordinates. Then, in view of the discussion in the previous section, we immediately reduce the equation to the memoryfree Markovian limit, while keeping the multidimensional formulation, and check the ﬂuctuation–dissipation balance. Subsequently we reduce also the spatial complexity to obtain the simple Langevin equation.

8.6.1 Langevin dynamics in generalized coordinates Consider a full Hamiltonian dynamical system with n degrees of freedom, expressed in 2n generalized coordinates and momenta z = {q, p}. The momenta are connected to the coordinates by the n × n mass tensor M (see (15.16) on page 401): ˙ p = Mq,

(8.38)

q˙ = M−1 p.

(8.39)

with inverse

The coordinates are distinguished in relevant coordinates q and irrelevant coordinates q . We partition the inverse mass tensor (as is done in the discussion on constraints, Section 17.9.3 on page 501) as X Y −1 , (8.40) M = YT Z so that

(

q˙

q˙

)=

X Y YT Z

p p

.

(8.41)

The next step is to ﬁnd a Langevin equation of motion for q by averaging over a canonical distribution for the double-primed subsystem: q˙ = X p + Yp , ∂V p˙ = − + friction + noise. ∂q

(8.42) (8.43)

264

Stochastic dynamics: reducing degrees of freedom

The canonical averaging is deﬁned as A(z) exp[−βH(z)] dz def . A = exp[−βH(z)] dz

(8.44)

We recognize the r.h.s. of (8.43) as the Langevin force, similar to the cartesian Langevin force of (8.1), but the l.h.s. is not equal to the simple mi v˙ i of the cartesian case. Instead we have, in matrix notation, and denoting q˙ by v, t ∂V mf d X−1 − v(t) = − ζ(τ )v(t − τ ) dτ + η(t). (8.45) dt ∂q 0 Here we have omitted the second term in (8.42) because it is an odd function of p that vanishes on averaging. In principle, X can be a function of primed coordinates, in which case the equations become diﬃcult to solve. But practice is often permissive, as we shall see. Let us have a closer look at the matrix X. Its elements are (Fixman, 1979) 1 ∂q ∂q k · l. (8.46) Xkl = mi ∂r i ∂r i i

Typical “relevant” degrees of freedom are cartesian coordinates of “superatoms” that represents a cluster of real atoms: the radius vector of the center of mass of a cluster of atoms, some linear combination of cartesian coordinates that represent collective motions (principal modes or principal components of a ﬂuctuation matrix), etc. Other cases (e.g., reaction coordinates) may involve distances between two particles or between two groups of particles. The inverse mass matrix X is particularly simple in these cases. For example, if the relevant coordinates are components of vectors Rk = i αki r i , the inverse mass tensor is diagonal with constant terms Xkl = δkl

1 α2 . mi ki

(8.47)

i

In the case that the relevant degree of freedom is a distance r12 between two particles with mass m1 and m2 , the inverse mass tensor has one element equal to (1/m1 ) + (1/m2 ), which is the inverse of the reduced mass of the two particles. The evaluation is equivalent to the evaluation in the case of constraints, treated in Section 17.9.3; see (17.201) on page 502. In all these cases the inverse mass tensor is constant and does not depend on timedependent coordinates. We shall from hereon restrict ourselves to such cases

8.6 Langevin dynamics

265

and write M for the inverse of X, yielding the general Langevin equation:11 t s Mv˙ = F (q ) − ζ(τ )v(t − τ ) dτ + η(t), (8.48) 0

˙ v = q, −1

M = X

(8.49) .

(8.50)

For X see (8.46). Of course, before applying this equation the user should check that the inverse mass tensor is indeed time-independent. For degrees of freedom involving angles this may not always be the case. We note that the formulation given in (8.48) includes the simple case that the relevant degrees of freedom are the cartesian coordinates of selected particles; the matrix M is then simply the diagonal matrix of particle masses. 8.6.2 Markovian Langevin dynamics We shall now consider the dissipation–ﬂuctuation balance for the case of generalized coordinates including a mass tensor. But since we cannot guarantee the validity of the second dissipation–ﬂuctuation theorem for the timegeneralized equation (8.48), we shall restrict ourselves to the Markovian multidimensional Langevin equation Mv˙ = Fs (q ) − ζv(t) + η(t).

(8.51)

Here ζ is the friction tensor. Dissipation–ﬂuctuation balance Consider the generalized equipartition theorem, treated in Section 17.10 and especially the velocity correlation expressed in (8.16) on page 503: vvT = M−1 kB T.

(8.52)

This equation is valid for the primed subsystem, where M is the inverse of X as deﬁned above. In order to establish the relation between friction and noise we follow the arguments of the previous section, leading to (8.35) on page 262, but now for the multidimensional case. The change due to friction is given by d ˙ T + vv˙ T = 2vv ˙ T vvT = vv dt = −2M−1 ζvvT . 11

(8.53)

The notation M should not cause confusion with the same notation used for the mass tensor of the full system, which would be obviously meaningless in this equation. Note that M used here in the Langevin equation is not a submatrix of the mass tensor of the full system! It should be computed as the inverse of X.

266

Stochastic dynamics: reducing degrees of freedom

In the ﬁrst line we have used the symmetry of vvT . The change in a step Δt due to noise is given by t+Δt t+Δt T −1 T −1 Δvv = M η(t ) dt η (t ) dt M t

t

= M−1 AM−1 Δt.

(8.54)

The matrix A is the noise correlation matrix: ηi (t0 )ηj (t0 + t) = Aij δ(t).

(8.55)

We used the symmetry of M−1 . Balancing the changes due to friction and noise, it is seen that friction and noise are related by A = 2kB T ζ.

(8.56)

This is the multidimensional analog of (8.35). It appears that the noise terms for the diﬀerent degrees of freedom are not independent of each other when the friction tensor is not diagonal, i.e., when the velocity of one degree of freedom inﬂuences the friction that another degree of freedom undergoes. We see from (8.56) that the friction tensor must be symmetric, as the Markovian noise correlation is symmetric by construction. It is also possible, and for practical simulations more convenient, to express the noise forces as linear combinations of independent normalized white noise functions ηk0 (t) with the properties ηk0 (t) = 0, ηk0 (t0 )ηl0 (t0 ηi (t) =

(8.57)

+ t) = δkl δ(t),

(8.58)

Bik ηk0 (t) or η = Bη 0 .

(8.59)

k

It now follows that BBT = A = 2ζkB T.

(8.60)

In order to construct the noise realizations in a simulation, the matrix B must be solved from this equation, knowing the friction matrix. The solution of this square-root operation is not unique; a lower trangular matrix is obtained by Choleski decomposition (see Engeln-M¨ ullges and Uhlig, 1996, for algorithms). Simple Langevin dynamics The generalized or the Markovian Langevin equation can be further approximated if the assumption can be made that the friction acts locally on each

8.6 Langevin dynamics

267

degree of freedom without mutual inﬂuence. In that case the friction tensor is diagonal and the simple Langevin equation is obtained: ˙ i = Fis − ζi vi (t) + η(t), (Mv)

(8.61)

ηi (t0 )ηj (t0 + t) = 2kB T ζi δij δ(t).

(8.62)

with

Although there is no frictional coupling, these equations are still coupled if the mass tensor M is not diagonal. In the common diagonal case the l.h.s. is replaced by ˙ i = mi vi . (Mv)

(8.63)

In the ultimate simpliﬁcation with negligible systematic force, as applies to a solute particle in a dilute solution, the simple pure Langevin equation is obtained: mv˙ = −ζv + η(t).

(8.64)

As this equation can be exactly integrated, the properties of v can be calculated; they serve as illustration how friction and noise inﬂuence the velocity, but are not strictly valid when there are systematic forces as well. The solution is 1 t −ζτ /m −ζt/m v(t) = v(0)e + e η(t − τ ) dτ, (8.65) m 0 which, after a suﬃciently long time, when the inﬂuence of the initial velocity has died out, reduces to 1 ∞ −ζτ /m e η(t − τ ) dτ. (8.66) v(t) = m 0 We see from (8.65) that in the absence of noise the velocity decays exponentially with time constant m/ζ, and we expect from the ﬁrst dissipation– ﬂuctuation theorem (page 259) that the velocity autocorrelation function will have the same exponential decay. This can be shown directly from (8.66) in the case that η(t) is a white noise with intensity 2ζkB T . It follows that, if η(t) is stationary, v(t) is also stationary; when the noise intensity is 2ζkB T , the variance of the velocity is kB T /m. Note that it is not necessary to specify the distribution function of the random variable. We can also compute the probability distribution ρ(v) for the velocity when equilibrium has been reached. To do this we need an equation for ρ(v, t) as it is generated by the stochastic process deﬁned by (8.64). Such

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Stochastic dynamics: reducing degrees of freedom

equations are Fokker–Planck equations,12 of which we shall see more examples in the following section. In this one-dimensional case the Fokker–Planck equation is ζ ∂ ζkB T ∂ 2 ρ ∂ρ = (ρv) + . (8.67) ∂t m ∂v m2 ∂t2 The equation is an expression of the conservation of total probability, leading to a continuum equation ∂ρ/∂t = −∇v (J), where J is the probability ﬂux consisting of a drift term due to friction and a diﬀusional term due to noise. The diﬀusional term follows from the fact that 1 ∂ρ ∂ρ = B (8.68) ∂t 2 ∂t implies that (see exercise 8.3) (Δv)2 = BΔt

(8.69)

with the variance of the velocity ﬂuctuation given by (2ζkB T /m2 )Δt (see (8.31), (8.32) and (8.35) on page 262). The equilibrium case (∂ρ/∂t = 0) has the solution & mv 2 m exp − , (8.70) ρ(v) = 2πkB T 2kB T which is the Maxwell distribution.

8.7 Brownian dynamics If systematic forces are slow, i.e., when they do not change much on the time scale τc = m/ζ of the velocity correlation function, we can average the Langevin equation over a time Δt > τc . The average over the inertial term Mv˙ becomes small and can be neglected; as a result the acceleration no longer ﬁgures in the equation. We obtain non-inertial dynamical equations: 0 ≈ Fi [q(t)] − ζij vj (t) + ηi (t), (8.71) j

or, in matrix notation: ζv = F + η(t),

(8.72)

yielding the Brownian equation for the velocities: v = q˙ = ζ −1 F + ζ −1 Bη 0 (t), 12

(8.73)

See van Kampen (1981) for an extensive treatment of the relation between stochastic equations and the corresponding Fokker–Planck equations.

8.8 Probability distributions and Fokker–Planck equations

BBT = 2ζkB T , η 0 (t)

= 0,

0 T

η (t0 )(η ) (t0 + t) = 1 δ(t). 0

269

(8.74) (8.75) (8.76)

In simulations the velocity can be eliminated and the positions can be updated by a simple Euler step: √ (8.77) q(t + Δt) = q(t) + ζ −1 F(t)Δt + ζ −1 Br Δt, where r is a vector of random numbers, each drawn independently from a probability distribution (conveniently, but not necessarily, Gaussian) with r = 0, r2 = 1.

(8.78)

Note that – as the dynamics is non-inertial – the mass does not enter in the dynamics of the system anymore. Apart from coupling through the forces, the coupling between degrees of freedom enters only through mutual friction coeﬃcients. For the simple Brownian dynamics with diagonal friction matrix, and using the diﬀusion constant Di = kB T /ζii , this equation reduces to qi (t + Δt) = qi (t) +

D Fi (t)Δt + ξ, kB T

(8.79)

where ξ is a random number, drawn from a probability distribution with ξ = 0, ξ 2 = 2DΔt.

(8.80)

One can devise more sophisticated forms that use the forces at half steps in order to integrate the drift part of the displacement to a higher order, but the noise term tends to destroy any higher-order accuracy. ¿From (8.77) it is seen that friction scales the time: decreasing the friction (or increasing the diﬀusion constant) has the same eﬀect as increasing the time step. It is also seen that the displacement due to the force is proportional to the time step, but the displacement due to noise is proportional to the square root of the time step. This means that slow processes that allow longer time steps are subjected to smaller noise intensities. For macroscopic averages the noise will eventually become negligible.

8.8 Probability distributions and Fokker–Planck equations In Section 8.6 we used a Fokker–Planck equation to derive the probability distribution for the velocity in the case of the simple pure Langevin equation (see page 267). This led to the satisfactory conclusion that the simple pure

270

Stochastic dynamics: reducing degrees of freedom

Langevin equation leads to a Maxwellian distribution. In this section we formulate Fokker–Planck equations for the more general Markovian Langevin equation and for the Brownian dynamics equation. What we wish to gain from the corresponding Fokker–Planck equations is insight into the steadystate and equilibrium behavior in order to judge their compatibility with statistical mechanics, and possibly also to obtain diﬀerential equations that can be solved analytically. Stochastic equations generate random processes whose distribution functions behave in time according to certain second-order partial diﬀerential equations, called Fokker–Planck equations. They follow from the master equation that describes the transition probabilities of the stochastic process. The Fokker–Planck equation is similar to the Liouville equation in statistical mechanics that describes the evolution of density in phase space resulting from a set of equations of motion; the essential diﬀerence is the stochastic nature of the underlying process in the case of Fokker–Planck equations.

8.8.1 General Fokker–Planck equations We ﬁrst give the general equations, and apply these to our special cases. Consider a vector of variables x generated by a stochastic equation:13 ˙ x(t) = a(x(t)) + Bη 0 (t),

(8.81)

with η 0 (t) independent normalized white noise processes, as speciﬁed by (8.58). The variables may be any observable, as coordinates or velocities or both. The ﬁrst term is a drift term and the second a diﬀusion term. The corresponding Fokker–Planck equation in the Itˆ o interpretation for the distribution function ρ(x, t) (van Kampen, 1981; Risken, 1989) is in matrix notation ∂ρ 1 T = −∇T tr (∇x ∇T x (aρ) + x BB ρ), ∂t 2

(8.82)

or for clarity written in components: ∂ 1 ∂2 ∂ρ =− (ai ρ) + Bik Bjk ρ. ∂t ∂xi 2 ∂xi ∂xj i

13

ij

(8.83)

k

We made a remark on this mathematically incorrect form of a stochastic diﬀerential equation on page 253 in relation to (8.4). The proper equation is dx = a dt + B dw, where w is a vector of Wiener processes.

8.8 Probability distributions and Fokker–Planck equations

271

8.8.2 Application to generalized Langevin dynamics Let us now apply this general equation to the general Markovian Langevin equation (8.51): q˙ = v,

(8.84) −1

v˙ = M

−1

F(q) − M

−1

ζv + M

Bη 0 .

(8.85)

The single-column matrix x consists of a concatenation of q and v. The single-column matrix a then consists of a concatenation of v and M−1 F(q)− M−1 ζv. Carefully applying (8.82) to this x (and assuming B to be constant) yields ∂ρ = −vT ∇q ρ − FT M−1 ∇v ρ + tr (M−1 ζ)ρ ∂t 1 +vT ζM−1 ∇v ρ + tr (M−1 BBT M−1 ∇q ∇q ρ). 2

(8.86)

Note that the noise coeﬃcient is related to the friction tensor by (8.60) on page 266: BBT = 2ζkB T.

(8.87)

This rather awesome multidimensional equation can of course be solved numerically and will give the same results as a simulation of the original stochastic equation. More insight is obtained when we reduce this equation to one dimension and obtain the rather famous Kramers equation (Kramers, 1940):14 ∂ρ F ∂ρ ρv ∂ρ ζ ζkB T ∂ 2 ρ ∂ρ = −v − + + ρ+ . ∂t ∂q m ∂v m ∂v m m2 ∂v 2

(8.88)

Even this much simpler equation cannot be solved analytically, but it can be well approximated to obtain classical rates for barrier-crossing processes. Kramer’s theory has been used extensively to ﬁnd damping corrections to the reaction rates derived from Eyring’s transition state theory. It is easy to ﬁnd the equilibrium distribution ρeq (q, v) by setting ∂ρ/∂t = 0 (see Exercise 8.5):

V (q) mv 2 exp − , (8.89) ρeq (q, v) ∝ exp − 2kB T kB T where V is the potential deﬁned by F = −dV /dq. Again, this is a satisfactory result compatible with the canonical distribution. 14

A generalization to colored noise and friction and with external noise has been given by Banik et al. (2000).

272

Stochastic dynamics: reducing degrees of freedom

8.8.3 Application to Brownian dynamics For Brownian dynamics the stochastic equations (8.73) and (8.74) are a function of q only. The corresponding Fokker–Planck equation is ∂ ∂ρ ∂2ρ =− [(ζ −1 F)i ρ] + kB T (ζ −1 )ij . ∂t ∂qi ∂qi ∂qj i

(8.90)

ij

For the case of diagonal friction which reduces to a set of one-dimensional equations (only coupled through the forces), the stochastic equation and the corresponding Fokker–Planck equation read D (F + Bη 0 ), kB T D ∂ ∂2ρ ∂ρ = − (ρF ) + D 2 . ∂t kB T ∂q ∂q q˙ =

(8.91) (8.92)

Setting ∂ρ/∂t = 0 and writing F = −dV /dq, we ﬁnd the equilibrium solution

V (q) , (8.93) ρ(q) ∝ exp − kB T which again is the canonical distribution. In order to obtain the canonical distribution by simulation using the stochastic Brownian equation, it is necessary to take the time step small enough for F Δt to be a good approximation for the step made in the potential V . If that is not the case, integration errors will produce deviating distributions. However, by applying an acceptance/rejection criterion to a Brownian step, a canonical distribution can be enforced. This is the subject of the following section.

8.9 Smart Monte Carlo methods The original Metropolis Monte Carlo procedure (Metropolis et al., 1953) consists of a random step in conﬁguration space, followed by an acceptance criterion ensuring that the accepted conﬁgurations sample a prescribed distribution function. For example, assume we wish to generate an ensemble with canonical probabilities: w(r) ∝ e−βV (r ) .

(8.94)

Consider a random conﬁgurational step from r to r = r + Δr and let the potential energies be given by E = V (r),

(8.95)

E = V (r ).

(8.96)

8.9 Smart Monte Carlo methods

273

The random step may concern just one coordinate or one particle at a time, or involve all particles at once. The sampling must be homogeneous over space. The transition probabilities W→ from r to r and W← from r to r should fulﬁll the detailed balance condition: w(r)W→ = w(r )W← ,

(8.97)

W→ w(r ) = e−β(E −E) . = W← w(r)

(8.98)

leading to the ratio

This is accomplished by accepting the step with a probability pacc → : E − E ≤ 0 : W→ = pacc → = 1,

for

E − E > 0 : W→ =

for

pacc →

−β(E −E)

=e

(8.99) ,

(8.100)

as is easily seen by considering the backward transition probability: for for

E − E < 0 : W← = eβ(E −E) ,

E − E ≥ 0 : W← = 1,

(8.101) (8.102)

which fulﬁlls (8.98). The acceptance with a given probability pacc → < 1 is realized by drawing a uniform random number 0 ≤ η < 1 and accepting the step when η < pacc → . When a step is not accepted, the previous step should be counted again. In the “smart Monte Carlo” procedure, proposed by Rossky et al. (1978), a Brownian dynamic step is attempted according to (8.79) and (8.80), sampling ξ from a Gaussian distribution. We denote the conﬁguration, force and potential energy before the attempted step by r, F and E and after the attempted step by r , F and E : r = r + βDΔtF + ξ.

(8.103)

The transition probability is not uniform in this case, because of the bias introduced by the force: (r − r − βDΔtF )2 pacc (8.104) W→ ∝ exp − → , 4DΔt because this is the probability that the random variable ξ is chosen such that this particular step results. Now imposing the detailed balance condition (8.98): W→ = e−β(E −E) , (8.105) W←

274

Stochastic dynamics: reducing degrees of freedom

we ﬁnd for the forward/backward acceptance ratio:

(r − r − βDΔtF )2 − (r − r − βDΔtF )2 pacc → = exp −β(E − E) + pacc 4DΔt ← = e−βΔ ,

(8.106)

with 1 1 Δ = E − E + (r − r) · (F + F ) − βDΔt(F 2 − F 2 ). 2 4

(8.107)

Note that Δ for the forward and backward step are equal in magnitude and opposite in sign. The acceptance is realized, similar to (8.98) and (8.100), by choosing: for for

Δ ≤ 0 : pacc → = 1, Δ>0:

pacc →

−βΔ

=e

(8.108) .

(8.109)

The latter acceptance is implemented by accepting the step when a homogeneous random number 0 ≤ η < 1 is smaller than exp(−βΔ). When a step is not accepted, the previous step should be counted again. The rejection of a step does destroy the dynamical continuity of the Brownian simulation, but ensures that the proper canonical distribution will be obtained. In practice, the time step – or rather the product DΔt – can be chosen such that almost all steps are accepted and the dynamics remains valid, at least within the approximations that have led to the Brownian stochastic equation.

8.10 How to obtain the friction tensor How can the friction tensor – or, equivalently, the noise correlation matrix – be obtained for use in Langevin or Brownian simulations? There are essentially three diﬀerent routes to obtain the friction tensor: (i) from theoretical considerations, (ii) from empirical data, (iii) from detailed MD simulations. The route to be chosen depends on the system and on the choice of ”irrelevant” degrees of freedom over which averaging should take place. In general the accuracy required for the friction tensor is not very high: it only inﬂuences the dynamical behavior of the system but not the thermodynamic equilibria. This is seen from the Fokker–Planck equation that appears to

8.10 How to obtain the friction tensor

275

yield a canonical distribution in conﬁguration space, even for the rather inaccurate Brownian dynamics, which is independent of the applied friction coeﬃcients as long as the ﬂuctuation–dissipation balance is maintained. It is likely that slow processes on a time scale much longer than the characteristic time scale of the friction, which is around m/ζ, will also be handled with reasonable accuracy. In many applications one is more interested in obtaining a fast sampling of conﬁguration phase than in accurately reproducing the real dynamics; in such cases one may choose a rather low friction in order to obtain a faster dynamical behavior. Ultimately one may choose not to add any friction or noise at all and obtain a fast dynamic sampling by just simulating Hamiltonian molecular dynamics of a reduced system with a proper potential of mean force. This is a quite common procedure in simulations based on ”superatoms.” In the following we consider a few examples of friction tensors.

8.10.1 Solute molecules in a solvent The most straightforward application of stochastic dynamics is the simulation of solute molecules in a solvent. In a dilute solution the friction is determined solely by the diﬀerence between the velocity v of the solute particle and the bulk velocity u of the solvent: F fr = −ζ(v − u),

(8.110)

and the friction tensor can at most be a 3 × 3 matrix for a non-spherical particle. For a spherical particle the friction tensor must be isotropic and equal to ζ1. We have not introduced terms like ”bulk velocities” of the bath particles before, implying that such velocities are assumed to be zero. Langevin and Brownian dynamics do not conserve momentum (and conserve energy only as an average) and should not be applied in the formulation given here when the application requires momentum and/or energy conservation. The friction coeﬃcient ζ follows from the diﬀusion coeﬃcient D of the particle and the temperature by the Einstein relation kB T . (8.111) D D can be obtained from experiment or from a simulation that includes the full solvent. The friction coeﬃcient can also be obtained from hydrodynamics if the solvent can be approximated by a continuum with viscosity η, yielding Stokes’ law for a spherical particle with radius a: ζ=

ζ = 6πηa.

(8.112)

276

Stochastic dynamics: reducing degrees of freedom

When the solution is not dilute, the most important addition is an interaction term in the systematic force; this can be obtained by thermodynamic integration from detailed simulations with pairs of particles at a series of constrained distances. But the friction force on solute particle i will also be inﬂuenced by the velocity of nearby solute particles j. This inﬂuence is exerted through the intervening ﬂuid and is called the hydrodynamic interaction. It can be evaluated from the Navier–Stokes equations for ﬂuid dynamics. The hydrodynamic interaction is a long-range eﬀect that decays with the inverse distance between the particles. The 1/r term in the interaction, averaged over orientations, is expressed as a mobility matrix, which forms the interaction part of the inverse of the friction matrix; this is known as the Oseen tensor. The equations are ζ −1 = H, 1 , Hii = 6πηa rrT 1 Hij = 1+ 2 , 8πηr r

(8.113) (8.114) (8.115)

where r = ri − rj and r = |r|. Each element of H, deﬁned above, is a 3 × 3 cartesian matrix; i, j number the solute particles. Hydrodynamic interactions are often included in stochastic modelling of polymers in solution, where the polymer is modelled as a string of beads and the solution is not modelled explicitly. Meiners and Quake (1999) have compared diﬀusion measurements on colloidal particles with Brownian simulations using the Oseen tensor and found excellent agreement for the positional correlation functions.

8.10.2 Friction from simulation In cases where theoretical models and empirical data are unavailable the friction parameter can be obtained from analysis of the “observed” forces in constrained simulations with atomic detail. If detailed simulations are done with the “relevant” degrees of freedom q constrained, the forces acting on the constrained degrees of freedom are the forces from the double-primed subsystem and – if carried to equilibrium – will approximate the sum of the systematic force and the random force that appear in the Langevin equation. The friction force itself will not appear as there are no velocities in the primed coordinates. The average of the constraint force F c will be the systematic force, which on integration will produce the potential of mean force. The ﬂuctuation ΔF c (t) will be a realization of the random force. If

Exercises

277

the second ﬂuctuation–dissipation theorem (8.26) holds, then ΔF c (t0 )ΔF c (t0 + t) = kB T ζ(t).

(8.116)

However, we have simpliﬁed the noise correlation function to a δ-function and the friction to a constant, which implies that ∞ ∞ 1 ζ(t) dt = ΔF c (t0 )ΔF c (t0 + t) dt. (8.117) ζ= kB T 0 0 One may also deﬁne the friction in terms of the diﬀusion constant D = kB T /ζ, so that (kB T )2 . (8.118) D = ∞ c c 0 ΔF (t0 )ΔF (t0 + t) dt In the multidimensional case, the cross correlation matrix of the constraint forces will similarly lead to the friction tensor.

Exercises 8.1 8.2 8.3 8.4 8.5

Solve mv˙ = −ζv + η(t) for the velocity v, given the velocity at t = 0, to yield (8.65). Compute v 2 (t) when friction and noise are switched on at t = 0 by taking the square of (8.65). Show that (8.69) follows from (8.69). Do this by showing that the time derivative of (Δv)2 equals B. Write (8.86) out in components. Find the equilibrium solution for the Kramers equation (8.88) by separating variables, considering ρ as a product of f (q) and g(v). This splits the equation; ﬁrst solve for the g(v) part and insert the result into the f (q) part.

9 Coarse graining from particles to ﬂuid dynamics

9.1 Introduction In this chapter we shall set out to average a system of particles over space and obtain equations for the variables averaged over space. We consider a Hamiltonian system (although we shall allow for the presence of an external force, such as a gravitational force, that has its source outside the system), and – for simplicity – consider a single-component ﬂuid with isotropic behavior. The latter condition is not essential, but allows us to simplify notations by saving on extra indexes and higher-order tensors that would cause unnecessary distraction from the main topic. The restriction to a single component is for simplicity also, and we shall later look at multicomponent systems. By averaging over space we expect to arrive at the equations of ﬂuid dynamics. These equations describe the motion of ﬂuid elements and are based on the conservation of mass, momentum and energy. They do not describe any atomic details and assume that the ﬂuid is in local equilibrium, so that an equation of state can be applied to relate local thermodynamic quantities as density, pressure and temperature. This presupposes that such thermodynamic quantities can be locally deﬁned to begin with. For systems that are locally homogeneous and have only very small gradients of thermodynamic parameters, averaging can be done over very large numbers of particles. For the limit of averaging over an inﬁnite number of particles, thermodynamic quantities can be meaningfully deﬁned and we expect the macroscopic equation to become exact. However, if the spatial averaging procedure concerns a limited number of particles, thermodynamic quantities need to be deﬁned also in terms of spatial averages and we expect the macroscopic equations to be only approximately valid and contain unpredictable noise terms. The situation is quite comparable to the averaging over “unimportant 279

280

Coarse graining from particles to ﬂuid dynamics

degrees of freedom” as was discussed in Chapter 8. The “important” degrees of freedom are now the density ρ(r) as a function of space, which is described with a limited precision depending on the way the spatial averaging is carried out. All other degrees of freedom, i.e., the particle coordinates within the restriction of a given density distribution, form the “unimportant” degrees of freedom, over which proper ensemble-averaging must be done. The forces that determine the evolution of density with time consist of three types: (i) systematic forces, depending on the coarse-grained density distribution (and temperature) itself; (ii) frictional forces, depending on the coarse-grained velocities; (iii) random forces that make up the unpredictable diﬀerence between the exact forces and the systematic plus frictional forces. In analogy with the behavior of a system with a reduced number of degrees of freedom (Chapter 8), we expect the random force to become of relatively less importance when the spatial averaging concerns a larger number of particles, and, in fact, a decrease in standard deviation with the square root of that number. If the spatial averaging is characterized by a smoothing distance a, then the relative standard deviation of the noise in mechanical properties is expected to be proportional to a−3/2 . As an example of a speciﬁc type of coarse graining, we can consider to simplify the description of particle positions by a density on a cubic spatial grid with spacing a. Instead of na3 particles (where n is the number density of the particles) we now have one density value per grid cell. So we must sum mechanical properties over roughly na3 particles: correlated quantities will become proportional to a3 and the noise will be proportional to the square root of that value. In Section 9.3 more precise deﬁnitions will be given. There are three reasons for obtaining the macroscopic equations for the behavior of ﬂuids by a process of coarse graining: (i) The assumptions on which the macroscopic equations rest (as validity of local density, bulk ﬂuid velocity, and pressure) are made explicit. (ii) The limits of application of the macroscopic equations become clear and correction terms can be derived. (iii) The macroscopic equations valid as approximation for a system of real particles are also an approximation for a system of diﬀerent and larger particles if their interactions are appropriately chosen. Thus the macroscopic problem can be solved by dynamic simulation of a many-particle system with a much smaller number of particles, be

9.2 The macroscopic equations of ﬂuid dynamics

281

it at the expense of increased noise. This is the basis of dissipative particle dynamics described in Chapter 11. In Section 9.2 an overview is given of the macroscopic equations of ﬂuid dynamics. This is done both as a reminder and to set the stage and notation for the systematic derivation of the macroscopic equations from microscopic equations of motion of the constituent particles, given in Section 9.3. Note that in Section 9.3 the macroscopic quantities are properly deﬁned on the basis of particle properties; in the macroscopic theory these quantities (density, ﬂuid velocity, pressure, etc.) are not really deﬁned, and their existence and validity as spatially-dependent thermodynamic quantities is in most textbooks assumed without further discussion.

9.2 The macroscopic equations of ﬂuid dynamics Note on notation We shall use vector notation as usual, but in some cases (like the derivatives of tensors) confusion may arise on the exact meaning of compound quantities, and a notation using vector or tensor components gives more clarity. Where appropriate, we shall give either or both notations and indicate cartesian components by greek indexes α, β, . . . , with the understanding that summation is assumed over repeated indexes. Thus ∂vβ /∂xα is the αβ component of the tensor ∇v, but ∂vα /∂xα is the divergence of v: ∇ · v.

The principles of single-component ﬂuid dynamics are really simple. The macroscopic equations that describe ﬂuid behavior express the conservation of mass, momentum and energy. The force acting on a ﬂuid element is – in addition to an external force, if present – given by a thermodynamic force and a frictional force. The thermodynamic force is minus the gradient of the pressure, which is related to density and temperature by a locally valid equation of state, and the frictional force depends on velocity gradients. In addition there is heat conduction if temperature gradients exist. Since we assume perfect homogeneity, there is no noise. Our starting point is the assumption that at every position in space the bulk velocity u(r) of the ﬂuid is deﬁned. Time derivatives of local ﬂuid properties can be deﬁned in two ways: (i) as the partial derivative in a space-ﬁxed coordinate frame, written as ∂/∂t and often referred to as the Eulerian derivative; (ii) as the partial derivative in a coordinate frame that moves with the bulk ﬂuid velocity u, written as D/Dt and often referred to as the Lagrangian derivative or the material or substantive derivative.

282

Coarse graining from particles to ﬂuid dynamics

The latter is related to the former by ∂ D = + u · ∇, Dt ∂t

D ∂ ∂ = + uα . Dt ∂t ∂xα

(9.1)

Some equations (as Newton’s equation of motion) are simpler when material derivatives are used. The next most basic local quantity is the mass density ρ(r) indicating the mass per unit volume. It is only a precise quantity for locally homogeneous ﬂuids, i.e., ﬂuids with small gradients on the molecular scale, on which no real ﬂuid can be homogeneous). We now deﬁne the mass ﬂux density J (r) as the mass transported per unit time and per unit area (perpendicular to the ﬂow direction): J = ρu.

(9.2)

9.2.1 Conservation of mass The continuity equation expresses the conservation of mass: when there is a net ﬂow of mass out of a volume element, expressed (per unit of volume) as the divergence of the mass ﬂux density, the total amount of mass in the volume element decreases with the same amount: ∂ρ ∂ρ ∂Jα + ∇ · J = 0, + = 0. (9.3) ∂t ∂t ∂xα The continuity equation can also be expressed in terms of the material derivative (using the deﬁnition of J ): Dρ + ρ∇ · u = 0. (9.4) Dt ¿From this formulation we see immediately that for an incompressible ﬂuid, for which ρ must be constant if we follow the ﬂow of the liquid, Dρ/Dt = 0 and hence the divergence of the ﬂuid velocity must vanish: ∇·u=0

(incompressible ﬂuid).

(9.5)

9.2.2 The equation of motion Next we apply Newton’s law to the acceleration of a ﬂuid element: ρ

Du = f (r) = f int + f ext , Dt

(9.6)

where f (r) is the total force acting per unit volume on the ﬂuid at position r. The total force is composed of internal forces arising from interactions within

9.2 The macroscopic equations of ﬂuid dynamics

283

the system and external forces, arising from sources outside the system. Internal forces are the result of a pressure gradient, but can also represent friction forces due to the presence of gradients in the ﬂuid velocity (or shear rate). Both kinds of forces can be expressed as the divergence of a stress tensor σ:1 ∂σαβ f int = ∇ · σ, fαint = . (9.7) ∂xβ Thus Newton’s law reads Du = ρ Dt Duα = ρ Dt

∂u + ρ (u · ∇)u = ∇ · σ + f ext , ∂t ∂σαβ ∂uα ∂uα ρ + ρuβ = + fαext . ∂t ∂xβ ∂xβ

ρ

(9.8)

Before elaborating on the stress tensor, we will formulate the equations for momentum conservation.

9.2.3 Conservation of linear momentum The momentum density, or the amount of linear momentum per unit volume, deﬁned with respect to a ﬁxed coordinate system, is given by ρu. This is the same as the mass ﬂux density J (see (9.2)). Conservation of momentum means that – in the absence of external forces – the amount of linear momentum increases with time as a result of the net inﬂux of momentum, or – in other words – that the time derivative of the momentum density equals minus the divergence of the momentum ﬂux density. Since momentum density is a vector, the momentum ﬂux density must be a tensor. We call it Π. The momentum conservation is expressed by ∂ (ρu) = −∇ · Π, ∂t

∂ ∂ (ρuα ) = − Παβ . ∂t ∂xβ

(9.9)

This expression can be proved to be valid (see below) when the following deﬁnition of the momentum ﬂux density tensor is adopted: Παβ = −σαβ + ρuα uβ .

(9.10)

This deﬁnition makes sense. There are two contributions: momentum can change either because a force gives an acceleration, or because particles ﬂow in or out of a region. The momentum ﬂux density tensor element Παβ is 1

For a more detailed discussion of the stress tensor and its relation to pressure, see Chapter 17, Section 17.7

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Coarse graining from particles to ﬂuid dynamics

the α component of the outward ﬂow of momentum through a unit area perpendicular to the xβ axis. Proof ∂uα ∂ρ ∂ρuα = ρ + uα ∂t ∂t ∂t ∂ρuβ ∂uα Duα − ρuβ − uα = ρ Dt ∂xβ ∂xβ ∂σαβ ∂ ∂ = − (ρuα uβ ) = − Παβ . ∂xβ ∂xβ ∂xβ In the ﬁrst line we have used (9.1) and (9.3) and in the second line (9.8).

9.2.4 The stress tensor and the Navier–Stokes equation The stress tensor σ is (in an isotropic ﬂuid) composed of a diagonal pressure tensor and a symmetric viscous stress tensor τ : σαβ = −p δαβ + ταβ .

σ = −p1 + τ ,

(9.11)

In an isotropic Newtonian ﬂuid where viscous forces are assumed to be proportional to velocity gradients, the only possible form2 of the viscous stress tensor is 2 ∂uα ∂uβ + ζ − η δαβ ∇ · u. + (9.12) ταβ = η ∂xβ ∂xα 3 The tensor must be symmetric with ∂uα /∂xβ + ∂uβ /∂uα as oﬀ-diagonal elements, because these vanish for a uniform rotational motion without internal friction, for which u = ω × r (ω being the angular velocity). We can split the viscous stress tensor into a traceless, symmetric part and an isotropic part: ⎞ ⎛ ∂u ∂uy ∂ux ∂uz ∂ux 2 ∂xx − 23 ∇ · u ∂y + ∂x ∂z + ∂x ⎟ ⎜ ∂u ∂uy y ∂ux ∂uz τ = η ⎝ ∂u 2 ∂yy − 23 ∇ · u ⎠ ∂x + ∂y ∂z + ∂y ∂uz ∂x

⎛

+

∂ux ∂z

∂uz ∂y

⎞ 1 0 0 +ζ∇ · u ⎝ 0 1 0 ⎠ . 0 0 1 2

+

∂uy ∂z

2 z 2 ∂u ∂z − 3 ∇ · u

For a detailed derivation see, e.g., Landau and Lifschitz (1987).

(9.13)

9.2 The macroscopic equations of ﬂuid dynamics

285

There can be only two parameters: the shear viscosity coeﬃcient η related to shear stress and the bulk viscosity coeﬃcient ζ related to isotropic (compression) stress. For incompressible ﬂuids, with ∇·u = 0, the viscous stress tensor simpliﬁes to the following traceless tensor: ∂uα ∂uβ (incompressible). (9.14) + ταβ = η ∂xβ ∂xα For incompressible ﬂuids there is only one viscosity coeﬃcient. The divergence of the viscous stress tensor yields the viscous force. For space-independent coeﬃcients, the derivatives simplify considerably, and the viscous force is then given by 1 visc 2 f = ∇ · τ = η∇ u + ζ + η ∇(∇ · u), 3 2 ∂ταβ ∂ uβ 1 visc 2 = η∇ uα + ζ + η . (9.15) fα = ∂xβ 3 ∂xα ∂xβ Combining (9.8) and (9.15) we obtain the Navier–Stokes equation (which is therefore only valid for locally homogeneous Newtonian ﬂuids with constant viscosity coeﬃcients): ρ

Du ∂u = ρ + ρ(u · ∇)u = ∇ · σ + f ext Dt ∂t 1 2 = −∇p + η∇ u + ζ + η ∇(∇ · u) + f ext . 3

(9.16)

Note that for incompressible ﬂuids the equation simpliﬁes to ∂u 1 η + (u · ∇)u = − ∇p + ∇2 u + f ext (incompressible). ∂t ρ ρ

(9.17)

The viscosity occurs in this equation only as the quotient η/ρ, which is called the kinematic viscosity and usually indicated by the symbol ν.

9.2.5 The equation of state The Navier–Stokes equation (9.16) and the continuity equation (9.3) are not suﬃcient to solve, for example, the time dependence of the density and velocity ﬁelds for given boundary and initial conditions. What we need in addition is the relation between pressure and density, or, rather, the pressure changes that result from changes in density. Under the assumption of local

286

Coarse graining from particles to ﬂuid dynamics

thermodynamic equilibrium, the equation of state (EOS) relates pressure, density and temperature: f (ρ, p, T ) = 0.

(9.18)

We note that pressure does not depend on the ﬂuid velocity or its gradient: in the equation of motion (see (9.8) and (9.11)) the systematic pressure force has already been separated from the velocity-dependent friction forces, which are gradients of the viscous stress tensor τ . The equation of state expresses a relation between three thermodynamic variables, and not just pressure and density, and is therefore – without further restrictions – not suﬃcient to derive the pressure response to density changes. The further restriction we need is the assumption that the thermodynamic change is adiabatic, i.e., that the change does not involve simultaneous heat exchange with a thermal bath. In real physical systems contact with thermal baths can only be realized at boundaries and is thus incorporated in boundary conditions. There is one exception: in an environment with given temperature the system is in interaction with a radiation ﬁeld with a black-body distribution typical for that temperature and absorbs and emits radiation, ﬁnally leading to thermal equilibration with the radiation ﬁeld. We may, however, for most practical purposes safely assume that the rate of equilibration with the radiation ﬁeld is negligibly slow compared to thermal conduction within the system and over its boundaries. The adiabaticity assumption is therefore valid in most practical cases. In simulations, where unphysical heat baths may be invoked, the adiabaticity assumption may be artiﬁcially violated. For small changes, the adiabatic relation between pressure and density change is given by the adiabatic compressibility κs : 1 dρ κs = , (9.19) ρ dp S or

dp dρ

= S

1 . κs ρ

(9.20)

A special case is an ideal gas for which pV cp /cV remains constant under an adiabatic change. This implies that cp p dp . (9.21) = dρ S cV ρ For dense liquids the compressibility is so small that for many applications the ﬂuid can be considered as incompressible, and ρ taken as constant in a

9.2 The macroscopic equations of ﬂuid dynamics

287

coordinate system that moves with the ﬂuid. This means that the divergence of the ﬂuid velocity vanishes (see (9.5)) and the Navier–Stokes equation (9.16) simpliﬁes to (9.17). 9.2.6 Heat conduction and the conservation of energy Note on notation In this section we need thermodynamic quantities per unit mass of material. We use overlined symbols as notation for quantities per unit mass in order not to cause confusion with the same thermodynamic quantities used elsewhere (without overline) per mole of component. These are intensive thermodynamic properties; the corresponding extensive properties are denoted by capitals (temperature is an exception, being intensive and denoted by T ). For internal energy we use u, not to be confused with ﬂuid velocity u. We deﬁne the following quantities: (i) Internal energy per unit mass u. This is the sum of the kinetic energy due to random (thermal) velocities and the potential energy due to interactions within the system.3 It does not include the kinetic energy per unit mass 12 u2 as a result of the ﬂuid velocity u. It has SI units J/kg or m2 s−2 . (ii) Enthalpy per unit mass h = u + p/ρ [J/kg]. (iii) Entropy per unit mass s [J kg−1 K−1 ], (iv) Thermodynamic potential per unit mass μ = h − T s [J/kg].

Due to adiabatic changes, and to dissipation caused by frictional forces, heat will be locally produced or absorbed, and the temperature will not be homogeneous throughout the system. Temperature gradients will cause heat ﬂow by conduction, and this heat ﬂow must be incorporated into the total energy conservation. If it is assumed that the heat ﬂux J q (energy per unit of time ﬂowing through a unit area) is proportional to minus the temperature gradient, then J q = −λ∇T,

(9.22)

where λ is the heat conduction coeﬃcient. The energy per unit mass is given by u + 12 u2 + Φext , and hence the energy per unit volume is ρu + 12 ρu2 + ρΦext . Here Φext is the potential energy per unit mass in an external ﬁeld (such as gz for a constant gravitational ﬁeld in the −z-direction), which causes the external force per unit volume f ext = −ρ∇Φ(r). Note that the external force per unit volume is not equal to minus the gradient of ρΦ. The energy of a volume element (per unit volume) changes with time for several reasons: (i) Reversible work is done on the volume element (by the force due to pressure) when the density changes: (p/ρ)(∂ρ/∂t). 3

For a discussion on the locality of energy, see Section 17.7.

288

Coarse graining from particles to ﬂuid dynamics

(ii) Reversible work is done by external forces; however, this work goes at the expense of the potential energy that is included in the deﬁnition of the energy per unit volume, so that the energy per unit volume does not change. (iii) Energy is transported with the ﬂuid (kinetic energy due to ﬂuid velocity plus internal energy), when material ﬂows into the volume element: −(u + 12 u2 + Φext )∇ · (ρu). (iv) Heat is produced by irreversible transformation of kinetic energy into heat due to friction: −u · [η∇2 u + (ζ + 13 η)∇(∇ · u)]. (v) Heat ﬂows into the volume element due to conduction: ∇ · (λ∇T ). Summing up, this leads to the energy balance equation p ∂ρ 1 2 ∂ ext 1 2 ext − u+ u +Φ = ∇ · (ρu) ρu + 2 ρu + ρΦ ∂t ρ ∂t 2

1 2 (9.23) −u · η∇ u + ζ + η ∇(∇ · u) + ∇ · (λ∇T ). 3 This concludes the derivation of the ﬂuid dynamics equations based on the assumptions that local density and local ﬂuid velocity can be deﬁned, and local thermodynamical equilibrium is deﬁned and attained. In the next secion we return to a more realistic molecular basis.

9.3 Coarse graining in space In this section we consider a classical Hamiltonian system of N particles with masses mi , positions r i , and velocities v i , i = 1, . . . , N . The particles move under the inﬂuence of a conservative interaction potential V (r 1 , . . . , r N ) and may be subject to an external force F ext i , which is minus the gradient of a potential Φ(r) at the position r i . Instead of considering the individual particle trajectories, we wish to derive equations for quantities that are deﬁned as “local” averages over space of particle attributes. We seek to deﬁne the local averages in such a way that the averaged quantities fulﬁll equations that approximate as closely as possible the equations of continuum ﬂuid dynamics, as described in the previous section. Exact correspondence can only be expected when the averaging concerns an inﬁnite number of particles. For ﬁnite-size averaging we hope to obtain modiﬁcations of the ﬂuid dynamics equations that contain meaningful corrections and give insight into the eﬀects of ﬁnite particle size. The spatial averaging can be carried out in various ways, but the simplest is a linear convolution in space. As stated in the introduction of this chapter,

9.3 Coarse graining in space

289

we consider for simplicity an isotropic ﬂuid consisting of particles of one type only. Consider the number density of particles n(r). If the particles are point masses at positions r i , the number density consists of a number of δ-functions in space: 0

n (r) =

N

δ(r − r i ).

(9.24)

i=1

The coarse-grained number density is now deﬁned as n(r) =

N

w(r − r i ),

(9.25)

i=1

where w(r) is a weight function, with dimension of one over volume. We shall take the weight function to be isotropic: w(r), with the property that it decays fast enough with r for the integral over 3D space to exist. The function is normalized ∞

w(r) 4πr2 dr = 1.

(9.26)

0

This condition implies that the integral of the number density over a large volume approximates the number of particles within that volume. The weight function is not prescribed in detail, but it should present a smoothing over space, and preferably (but not necessarily) be positive and monotonically decreasing with r. A useful and practical example is the 3D Gaussian function √ −3 r2 w(r) = (σ 2π) exp − 2 . (9.27) 2σ Note on the symmetry of weight functions We have made the weight function w(r) a function of the distance only, and therefore the weight function is perfectly symmetric in space, and invariant for rotation. This is not a necessary condition, and we could take a weight function w(r) that is not rotationally invariant, but still of high, e.g., cubic, symmetry, such as a product function w(r) = w1 (x)w1 (y)w1 (z),

(9.28)

where w1 (x) is a symmetric function in x. Product functions have the advantage that their Fourier transforms are a product of the Fourier transforms of the onedimensional weight functions. Normalization according to (9.26) is not valid for product functions in general, but must be replaced by the normalization of each of the 1D functions: +∞ w1 (x) dx = 1. (9.29) −∞

Simple one-dimensional weight functions are listed below:

290

Coarse graining from particles to ﬂuid dynamics

(i) Constant weight w1 (x) = 1/(2a) for |x| ≤ a = 0 for |x| > a.

(9.30)

The Fourier transform of this function is a sinc function, sin ka/(ka). (ii) Triangular weight w1 (x) = a−2 (a − |x|) for |x| ≤ a = 0 for |x| > a.

(9.31)

This function is in fact a convolution of the previous function with itself, and therefore its Fourier transform is the square of a sinc function [2 sin( 12 ka)/(ka)]2 . (iii) Sinc function w1 (x)

1 sin(πx/a) . a πx/a

(9.32)

This function has a band-limited Fourier transform that is constant up to |k| = π/a and zero for larger |k|. (iv) Normal distribution x2 1 √ w1 (x) = exp − 2 . (9.33) 2σ σ 2π The Fourier transform of this function is a Gaussian function of k, proportional to exp(− 12 σ 2 k 2 ). The 3D product function is a Gaussian function of the distance r. In fact, the Gaussian function is the only 1D function that yields a fully isotropic 3D product function, and is therefore a preferred weight function.

9.3.1 Deﬁnitions We now deﬁne the following averaged quantities: (i) Number density def

n(r) =

w(r − r i ).

(9.34)

mi w(r − r i ).

(9.35)

i

(ii) Mass density def

ρ(r) =

i

(iii) Mass ﬂux density or momentum density def mi v i w(r − r i ). J (r) = i

(9.36)

9.3 Coarse graining in space

291

(iv) Fluid velocity def

u(r) =

J (r) . ρ(r)

(9.37)

This deﬁnition is only valid if ρ diﬀers from zero. The ﬂuid velocity is undetermined for regions of space where both the mass density and the mass ﬂux density are zero, e.g., outside the region to which the particles are conﬁned. (v) Force per unit volume def f (r) = F i w(r − r i ), (9.38) i

where F i is the force acting on particle i. This force consists of an internal contribution due to interactions between the particles of the system, and an external contribution due to external sources. (vi) Stress tensor and pressure The deﬁnitions of the stress tensor σ, the pressure, and the viscous stress tensor, are discussed below. (vii) Momentum ﬂux density tensor def Παβ (r) = −σαβ (r) + mi viα viβ w(r − r i ). (9.39) i

Note that the deﬁnition of Π uses the weighted particle velocities and not the ﬂuid velocities as in (9.10). With the present deﬁnition linear momentum is conserved, but Newton’s equation for the acceleration has extra terms (see below). (viii) Temperature 2 def i mi (v i − u(r)) w(r − r i ) T (r) = . (9.40) 3kB n(r) Temperature is only deﬁned for regions where the number density diﬀers from zero. It is assumed that all degrees of freedom behave classically so that the classical equipartition theorem applies. For hard quantum degrees of freedom or for holonomic constraints corrections must be made.

9.3.2 Stress tensor and pressure The coarse-grained stress tensor should be deﬁned such that its divergence equals the internal force per unit volume (see (9.7)). As is elaborated in

292

Coarse graining from particles to ﬂuid dynamics

Chapter 17 in connection with locality of the virial, there is no unique solution, because any divergence-free tensor can be added to the stress tensor without changing the force derived from it. For forces between point particles, the stress tensor is localized on force lines that begin and end on the particles, but are further arbitrary in shape. Schoﬁeld and Henderson (1982) have suggested the following realization of the stress tensor: int Fiα δ(r − r c ) dxcβ , (9.41) σαβ = − C0i

i

where the integral is taken over a path C0i starting at an arbitrary reference point r 0 and ending at r i . The generalization to a coarse-grained quantity is straightforward: the δ-function in (9.41) is replaced by the weight function w and the reference point is chosen at the position r. Thus we deﬁne the averaged stress tensor as def int Fiα w(r − r c ) dxcβ , (9.42) σαβ (r) = − Ci

i

where the integral is taken over a path Ci starting at r and ending at r i . It is logical to choose straight lines for the paths. The divergence of this stress tensor now yields the averaged internal force per unit volume, as deﬁned in (9.38):

Proof

(∇ · σ)(r) = f int (r).

(9.43)

∂ w(r − r c )dxcβ (∇ · σ)α = − ∂xβ Ci i ri ∂ int = Fiα w(r − r c )dxcβ r ∂xcβ i int = Fiα w(r − r i ).

(9.44)

int Fiα

i

9.3.3 Conservation of mass The mass conservation law of continuum mechanics (9.3): ∂ρ + ∇ · J = 0, ∂t

(9.45)

9.3 Coarse graining in space

293

is valid and exact for the averaged quantities. Proof Note that ρ (see (9.35)) is time dependent through the time dependence of r i , and that the gradient of w with respect to r i equals minus the gradient of w with respect to r: ∂ρ = − mi (∇w(r − r i )) · v i ∂t i = −∇ · mi v i w(r − r i ) i

= −∇ · J

9.3.4 Conservation of momentum The momentum conservation law of continuum mechanics (9.9): ∂ ∂ (ρuα ) = − Παβ ∂t ∂xβ

(9.46)

(valid in the absence of external forces) is valid and exact for the averaged quantities. Proof After applying (9.37) and (9.39) we must prove that, in the absence of external forces, ∂σαβ ∂Jα ∂ = − mi viα viβ w(r − r i ). ∂t ∂xβ ∂xβ

(9.47)

i

Filling in (9.36) on the l.h.s., we see that there are two time-dependent terms, viα and r i , that need to be diﬀerentiated: ∂w(r − r i ) ∂Jα = mi v˙ iα w(r − r i ) − mi viα viβ ∂t ∂xβ i i ∂ = fαint − mi viα viβ w(r − r i ). ∂xβ i

Since the divergence of σ equals f int (r) (see (9.43)), we recover the r.h.s. of (9.47).

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Coarse graining from particles to ﬂuid dynamics

9.3.5 The equation of motion The equation of motion of continuum mechanics (9.6): Du = f (r) (9.48) Dt now has a slightly diﬀerent form and contains an additional term. Working out the l.h.s. we obtain Duα ∂ρ ∂Jα ∂Jα ∂ρ ρ , (9.49) = + uβ + uβ − uα Dt ∂t ∂xβ ∂t ∂xβ ρ

and carrying through the diﬀerentiations, using (9.35) and (9.36), we ﬁnd ∂w(r − r i ) Duα = fα (r) − mi (viα − uα )(viβ − uβ ) ρ Dt ∂xβ i ∂ σαβ − = mi (viα − uα )(viβ − uβ )w(r − r i ) . (9.50) ∂xβ i

The step to the last equation follows since the terms with the partial derivatives ∂uα /∂xβ and ∂uβ /∂xβ vanish. For example: ∂uα ∂uα mi (viβ − uβ )w(r − r i ) = (Jβ − ρuβ ) = 0, ∂xβ ∂xβ i

because J = ρu. It thus turns out that there is an extra term in the ﬂuid force that is not present in the equation of motion of continuum mechanics. It has the form of minus the divergence of a tensor that represents the weighted particle velocity deviation from the ﬂuid velocity. This term is also exactly the diﬀerence between the particle-averaged momentum ﬂux density (9.39) and the momentum ﬂux density (9.10) as deﬁned in ﬂuid mechanics. Let us call this term the excess momentum ﬂux density Πexc : mi [viα − uα (r)][viβ − uβ (r)]w(r − r i ). (9.51) Πexc αβ (r) = i

Its divergence gives an extra force per unit volume. Inspection of this term shows that it represents the thermal kinetic energy density, with an equilibrium average determined by equipartition: Πexc αβ (r) = n(r)kB T (r)δαβ .

(9.52)

This term is indeed the missing term if we compare Πexc to the pressure computed from virial and kinetic energy in statistical mechanics (Chapter 17, (17.127) on page 485). It has no inﬂuence on the force unless there is a

9.4 Conclusion

295

gradient of number density or a gradient of temperature. In addition to the average contribution, Πexc has a ﬂuctuating component that adds noise to the pressure and to the force.

9.4 Conclusion As we have seen, coarse graining of a Hamiltonian ﬂuid by spatial averaging with a weight function, yields the conservation laws, if the macroscopic quantities are properly deﬁned. However, the equation of motion has an extra term that can be written as the divergence of an extra pressure term (9.52). It is related to the local thermal kinetic energy and equals the kinetic term required to describe pressure in statistical mechanics. With this term included, and including the local stress tensor derived from the virial of the local force (9.42), the pressure is a property of the system, determined by the density of particles and by the interactions between the particles. This is a manifestation of the local EOS. In ﬂuid dynamics, where the description in terms of interacting particles is lost, the EOS is an additional “property” of the system that enables the determination of local pressure based on density and temperature (or energy density or entropy density). Note that local forces between particle pairs, which contribute to the local momentum ﬂux density and therefore to the local pressure, cancel in the coarse-grained force density and do not play a direct role in ﬂuid forces. Another important diﬀerence between the dynamics of a system of interacting particles and a ﬂuid continuum is that the coarse-grained dynamical properties are averages over a ﬁnite number of particles and are therefore ﬂuctuating quantities with limited precision. This introduces “noise” and will have an inﬂuence on chaotic features of ﬂuid dynamics, as turbulence, but only when the length scale of such features approach molecular size ranges. For macroscopic length scales the averaging can be done over such a large number of particles that the ﬂuctuations become negligible. In the intermediate range, where details on an atomic scale are not needed but ﬂuctuations are not negligible, the term mesoscopic dynamics is used. Mesoscopic dynamics can be realized either with particles (as Langevin or Brownian dynamics with superatomic system description) or with continuum equations, for example on a grid.

Exercises 9.1

Derive (9.15) from (9.12).

296

9.2

Coarse graining from particles to ﬂuid dynamics

Derive the second line of (9.50) from the ﬁrst line. Note that also the ﬂuid velocity is a function of spatial coordinates.

10 Mesoscopic continuum dynamics

10.1 Introduction

The term “mesoscopic” is used for any method that treats nanoscale system details (say, 10 to 1000 nm) but averages over atomic details. Systems treated by mesoscopic methods are typically mixtures (e.g., of polymers or colloidal particles) that show self-organization on the nanometer scale. Mesoscopic behavior related to composition and interaction between constituents comes on top of dynamic behavior described by the macroscopic equations of ﬂuid dynamics; it is on a level between atoms and continuum ﬂuids. In mesoscopic dynamics the inherent noise is not negligible, as it is in macroscopic ﬂuid dynamics. Mesoscopic simulations can be realized both with particles and with continuum equations solved on a grid. In the latter case the continuum variables are densities of the species occurring in the system. Particle simulations with “superatoms” using Langevin or Brownian dynamics, as treated in Chapter 8, are already mesoscopic in nature but will not be considered in this chapter. Also the use of particles to describe continuum equations, as in dissipative particle dynamics described in Chapter 11, can be categorized as mesoscopic, but will not be treated in this chapter. Here we consider the continuum equations for multicomponent mesoscopic systems in the linear response approximation. The latter means that ﬂuxes are assumed to be linearly related to their driving forces. This, in fact, is equivalent to Brownian dynamics in which accelerations are averaged-out and average velocities are proportional to average, i.e., thermodynamic, forces. The starting point for mesoscopic dynamics will therefore be the irreversible thermodynamics in the linear regime, as treated in Chapter 16, Section 16.10. 297

298

Mesoscopic continuum dynamics

10.2 Connection to irreversible thermodynamics We start with the irreversible entropy production per unit volume σ of (16.98) on page 446. Replacing the “volume ﬂux” J v by the bulk velocity u we may write 1 1 1 1 σ = Jq · ∇ − u · ∇p + I · E − J i · (∇μi )p,T . (10.1) T T T T i

Here we recognize heat ﬂux J q and electric current density I, driven by a temperature gradient and an electric ﬁeld, respectively. The second term relates to the irreversible process of bulk ﬂow caused by a force density, which is the gradient of the (generalized) pressure tensor including the viscous stress tensor (see Section 9.2.4 on page 284). The last term is of interest for the relative diﬀusional ﬂux of particle species, driven by the gradient of the thermodynamic potential of that species. Any bulk ﬂow J i = ci u, with all species ﬂowing with the same average speed, does not contribute to this term since ci (∇μi )p,T = 0, (10.2) i

as a result of the Gibbs–Duhem relation. The term can be written as 1 Ji σdiﬀ = − − u · [ci (∇μi )p,T ]. (10.3) T ci i

The term J i /ci − u = udi denotes the average relative velocity of species i with respect to the bulk ﬂow velocity, and we may deﬁne the diﬀerence ﬂux J di as def

J di = ci udi = J i − ci u.

(10.4)

It is clear that there are only n−1 independent diﬀerence ﬂuxes for n species, and the sum may be restricted1 – eliminating species 0 (the “solvent”) – to species 1 to n − 1, which yields the equivalent form (see also Chapter 16, Eq. (16.104)): 1 Ji J0 σdiﬀ = − · [ci (∇μi )p,T ]. − (10.5) T ci c0 i

Simplifying to a two-component system, with components numbered 0 and 1, the diﬀusional entropy production can be written as 1 σdiﬀ = − [ud1 c1 (∇)μ1 )p,T + ud1 c1 (∇μ1 )p,T ], T 1

This is indicated by the prime in the sum.

(10.6)

10.2 Connection to irreversible thermodynamics

299

with the Gibbs–Duhem relation c1 (∇μ1 )p,T + c2 (∇)μ1 )p,T = 0, or alternatively as 1 σdiﬀ = − (u1 − u0 )c1 (∇μ1 )p,T . (10.7) T The linear response assumption is that a system that is not in overall equilibrium will develop ﬂows J i proportional to driving forces X j (deﬁned such that σ = i J i · X i ) according to the Onsager phenomenological relations (16.109) and (16.111): Lij X j ; Lij = Lji . (10.8) Ji = j

On the mesoscopic level of theory the transport coeﬃcients Lij are input parameters for mesoscopic simulations; they can be derived from experiment or from non-equilibrium simulations at the atomic level, but do not follow from mesoscopic system simulation. One may adopt the simplifying but poor assumption that there are only diagonal transport coeﬃcients. For a two-component system there is only one coeﬃcient connecting the relative particle ﬂux (i.e., diﬀusional ﬂux) to the chemical potential gradients. This coeﬃcient is related to the diﬀusion constant in the following way. For a dilute or ideal solution of component 1 in solvent 0 (i.e., small c1 ), the thermodynamic potential (see Chapter 16, Section 16.6 on page 435) is given by μ1 = μ01 + RT ln(c1 /c0 ),

(10.9)

and hence ∇μ1 =

RT ∇c1 , c1

(10.10)

while the diﬀusional ﬂux equals the diﬀusion constant D times the concentration gradient: J d1 = c1 (u1 − u0 ) = −D∇c1 .

(10.11)

Combined this implies that u1 − u0 = −

D ∇μ1 . RT

(10.12)

The negative gradient of μ1 is the thermodynamic force that tries to move component 1 with respect to component 0; in the steady state the thermodynamic force is counterbalanced by an average frictional force ζ(u1 − u0 ), where ζ is the friction coeﬃcient. The friction coeﬃcient is therefore related

300

Mesoscopic continuum dynamics

to the diﬀusion coeﬃcient by RT . (10.13) D For n-component mixtures there are n − 1 independent concentrations and 1 2 2 n(n−1) diﬀusion coeﬃcients. In the local coupling approximation (LCA) it is assumed that the transport coeﬃcient is proportional to the local density and the gradient of the thermodynamic potential. Now consider the time evolution of the concentration ci of species i. In the mesoscopic literature it is costumary to indicate this quantity by the density ρi , expressed either in number of particles or in moles per unit volume, and we shall adopt this convention. We shall focus on the structural rearrangements in mixtures following material transport and therefore simplify the system considerably by considering an isothermal/isobaric system, in which there is no heat ﬂux, electric current, or bulk ﬂow. The continuity equation for species i reads ∂ρi = −∇J i (10.14) ∂t with the ﬂux in the local coupling approximation and including a random due to thermal ﬂuctuation: term J rand i ζ=

, J i = −M ρi ∇μi + J rand i

(10.15)

where we take for simplicity a single transport coeﬃcient M=

D = ζ −1 RT

(10.16)

and where J rand is the random residual of the ﬂux which cannot be neglected i when the coarse-graining averages over a ﬁnite number of particles. This “noise” must satisfy the ﬂuctuation–dissipation theorem and is intimately linked with the friction term; it is considered in the next section. Note The friction can be treated with considerably more detail, e.g., one may distinguish the frictional contribution of diﬀerent species (if there are more than two species), in which case the ﬂux equation becomes a matrix equation. One may also generalize the local coupling approximation inherent in (10.15) and use a spread function for the local friction. So the general form is J i (r) = − Λij (r; r )∇μj (r ) dr + J rand , (10.17) i j 2

V

The mutual diﬀusion constants are complicated functions of the concentrations, but the dependencies become much simpler in the Maxwell–Stefan description in terms of inverse diﬀusion constants or friction coeﬃcients, because the frictional forces with respect to other components add up to compensate the thermodynamic force. See Wesselingh and Krishna (1990) for an educational introduction to the Maxwell–Stefan approach, as applied to chemical engineering.

10.3 The mean ﬁeld approach to the chemical potential

301

with Λij (r; r ) = M ρi δij δ(r − r )

(10.18)

in the local coupling approximation.

The equation for the evolution of the density of species i is given by the continuity equation for each species, provided there are no chemical reactions between species: ∂ρ1 = −∇ · J i = M ∇ · (ρi ∇μi ) − ∇J rand . i ∂t

(10.19)

10.3 The mean ﬁeld approach to the chemical potential What we are still missing is a description of the position-dependent chemical potential given the density distribution. When we have such a relation the gradients of the thermodynamic potentials are known and with a proper choice of the mobility matrix the time evolution of a given density distribution can be simulated. Thus we can see how an arbitrary, for example homogeneous, density distribution of, e.g., the components of a block copolymer, develops in time into an ordered structural arrangement. The thermodynamic potential is in fact a functional of the density distribution, and vice versa. In order to ﬁnd the chemical potential, one needs the total free energy A of the system, which follows in the usual way from the partition function. The Hamiltonian can be approximated as the sum of a local contribution, independent of the density distribution, based on a local description of the unperturbed polymer, and a non-local contribution resulting from the density distribution. Simple models like the Gaussian chain model suﬃce for the local contribution. The non-local contribution to the chemical potential due to the density distribution is in mesoscopic continuum theory evaluated in the mean-ﬁeld approximation, essentially following Landau–Ginzburg theory. If the free energy A, which is a functional of the density distribution, is known, the position-dependent chemical potential is its functional derivative to the density: δA μ(r) = . (10.20) δρ(r) When the system is in equilibrium, the density distribution is such that A is a global minimum, and the chemical potential is a constant. By adding an energy term U (r), which we call the “external ﬁeld”, to the Hamiltonian, the equilibrium density distribution will change; there is a bijective relation between the density distribution and the external ﬁeld U . The evaluation

302

Mesoscopic continuum dynamics

of the functionals is quite intricate and the reader is referred to the original literature: Fraaije (1993) and Fraaije et al. (1997). The theory has been applied to several di- and triblock-copolymer melts, such as the industrially important triblock polymer “pluronic” that consists of three consecutive blocks ethylene oxide – propylene oxide – ethylene oxide, e.g., EO13 -PO30 -EO13 . Spontaneous formation of lamellar, hexagonal, bicubic and other structures has been observed, where the order remains local and only very slowly extends to larger distances. When shear is applied, ordering over longer distances is induced. See Fig. 10.1 for an example.3

3

Some of the relevant articles are Zvelindovski (1998a, 1998b), van Vlimmeren et al. (1999), Maurits et al. (1998a, 1998b, 1999), Sevink et al. (1999) and Morozov et al. (2000).

10.3 The mean ﬁeld approach to the chemical potential

303

a

d

100

15,000

b

e

1,000

20,000

c

f

10,000

24,000

Figure 10.1 Six snapshots of the evolution of a diblock-copolymer melt of the type A10 B10 in a mesoscopic continuum simulation at T = 300 K. At time 0 a homogeneous melt is subjected to a repulsive A–B interaction (χ = 0.8); it develops a lamellar structure (a–c). After 10,000 time steps (c) a shear rate of 0.001 box lengths per time step is imposed; the lamellar structure orients itself along the direction of shear into a co-called perpendicular orientation (d–f). The dimensionless density of A is shown as shades of gray only for values larger than its volume averaged value (= 0.5). Figure courtesy of Dr Agur Sevink, Leiden University. See also Zvelindovsky et al. (1998a, 1998b).

11 Dissipative particle dynamics

In this chapter we consider how continuum dynamics, described by continuum equations that are themselves generalizations of systems of particles, can be described by particles again. The particle description in this case is not meant to be more precise than the continuum description and to represent the system in more detail, but is meant to provide an easier and more physically appealing way to solve the continuum equations. There is the additional advantage that multicomponent systems can be modeled, and by varying the relative repulsion between diﬀerent kinds of particles, phenomena like mixing and spinodal decomposition can be simulated as well. The particles represent lumps of ﬂuid, rather than speciﬁed clusters of real molecules, and their size depends primarily on the detail of the boundary conditions in the ﬂuid dynamics problem at hand. The size may vary from superatomic or nanometer size, e.g., for colloidal systems, to macroscopic size. Since usually many (millions of) particles are needed to ﬁll the required volume with suﬃcient detail, is it for eﬃciency reasons necessary that the interactions are described in a simple way and act over short distances only to keep the number of interactions low. Yet, the interactions should be suﬃciently versatile to allow independent parametrization of the main properties of the ﬂuid as density, compressibility and viscosity. Although dissipative particle dynamics (DPD), which is meant to represent continuum mechanics, diﬀers fundamentally from coarse-grained superatom models, which are meant to represent realistic molecular systems in a simpliﬁed way, the distinction in practice is rather vague and the term DPD is often also used for models of polymers that are closer to a superatom approach. The origin of DPD can be identiﬁed as a paper by Hoogerbrugge and Koelman (1992),1 who described a rather intuitive way of treating ﬂuid dynamics 1

See also Koelman and Hoogerbrugge (1993), who applied their method to the study of hardsphere suspensions under shear.

305

306

Dissipative particle dynamics

problems with particles. Essentially their model consists of particles with very simple, short-ranged conservative interactions with additional friction and noise terms that act pairwise and conserve momentum and average energy. The addition of friction and noise functions as a thermostat and allows an extra parameter to inﬂuence the viscosity of the model. But there are predecessors: notably the scaled particle hydrodynamics (SPH), reviewed by Monaghan (1988) with the aim to solve the equations of ﬂuid dynamics by the time evolution of a set of points. SPH was originally developed to solve problems in astrophysics (Lucy, 1977). It is largely through the eﬀorts of P. Espa˜ nol2 that DPD was placed on a ﬁrm theoretical footing, and resulted in a formulation where the equation of state (i.e., pressure and temperature as functions of density and entropy or energy) and properties such as viscosity and thermal conductivity can be used as input values in the model, rather than being determined by the choice of interparticle interactions (Espa˜ nol and Revenga, 2003). In Espa˜ nol’s formulation each particle has four attributes: position, momentum, mass and entropy, for which appropriate stochastic equations of motion are deﬁned.3 Another model, originated by Flekkøy and Coveney (1999),4 uses ﬂuid “particles” based on Voronoi tesselation that divides space systematically in polyhedral bodies attributed to moving points in space. We shall not describe these more complicated DPD implementations, but rather give a short description of a popular and simple implementation of DPD given by Groot and Warren (1997). This implementation is close to the original model of Hoogerbrugge and Koelman (1992). One should be aware that simplicity comes at a price: models of this kind have intrinsic properties determined by the interaction functions and their parameters and simulations are generally needed to set such properties to the desired values. The model contains stochastic noise and friction, and represents therefore a Langevin thermostat (see Chapter 6, page 196). Such a distributed thermostat causes isothermal behavior rather than the adiabatic response that is usually required in realistic ﬂuid dynamics. 2 3

4

Espa˜ nol (1995) derived hydrodynamic equations from DPD and evaluated the probability density from the Fokker–Planck equation corresponding to the stochastic equations of motions. The formal derivation of thermodynamically consistent ﬂuid particle models is based on the GENERIC (General Equation for Non-Equilibrium Reversible-Irreversible Coupling) formalism ¨ ¨ ¨ ¨ of Ottinger (Grmela and Ottinger, 1997; Ottinger and Grmela, 1997; Ottinger, 1998). In this formalism the change in a set of variables that characterize the state of a system is expressed in terms of the dependence of energy and entropy on the state variables; this is done in such a way that energy is conserved and entropy cannot decrease, while the ﬂuctuation–dissipation theorem is satisﬁed. See Espa˜ nol et al. (1999) for the application to hydrodynamic generalization. See also Flekkøy et al. (2000) and Espa˜ nol (1998). Serrano and Espa˜ nol (2001) elaborated on this model and the two approaches were compared by Serrano et al. (2002).

11.1 Representing continuum equations by particles

307

11.1 Representing continuum equations by particles The system consists of particles with mass mi , position r i and velocity v i . Each particle represents a ﬂuid element that moves coherently. The particles interact pairwise though two types of forces: a potential-derived conservative force and a dissipative friction force that depends on the velocity diﬀerence between two interacting particles. The energy dissipation due to the dissipative force is balanced by a random force, so that the total average kinetic energy from motion with respect to the local center of mass, excluding the collective kinetic energy (the “temperature”), remains constant. Since all forces act pairwise in the interparticle direction and are short-ranged, the sum of forces is zero and both linear and angular momentum is conserved, even on a local basis. Since mass, energy and momentum conservation are the basis of the continuum equations of ﬂuid dynamics, DPD dynamics will follow these equations on length scales larger than the average particle separation and on time scales larger than the time step used for integration the equations of motion. The equations of motion are Newtonian: r˙ i = v i v˙ i = F i =

(11.1) F ij ,

(11.2)

j =i

where D R F ij = F C ij + F ij + F ij .

(11.3)

The conservative force on particle i due to j is repulsive with a range 1 given by: FC ij = aij (1 − rij )

r ij rij < 1, rij

= 0 rij ≥ 1,

(11.4) (11.5)

with r ij = r i − r j . This corresponds to a quadratic repulsive potential with one parameter aij . Note that the distance is scaled such that the maximum interaction range equals 1. The dissipative force is given by F ij = −γwD (rij )(v ij · r ij )

r ij 2 . rij

(11.6)

It acts in the direction of r ij and is proportional to the component of the velocity diﬀerence in the interparticle direction, being repulsive when particles move towards each other and attractive when they move away. Thus

308

Dissipative particle dynamics

it damps the relative motion of the two particles. The parameter γ measures the strength of the damping; wD (rij ) is a weight function vanishing for rij > 1. The random force also acts in the interparticle direction: R FR ij = σw (rij )

r ij θij , rij

(11.7)

where σ is the strength, wR (rij ) a weight function vanishing for rij > 1, and θij a random function with average zero and with no memory: θ(0)θ(t) = δ(t), uncorrelated with the random function on any other particle pair. The distribution function of θ can be chosen to be normal, but that is not a requirement. Espa˜ nol and Warren (1995) showed that the ﬂuctuation– dissipation theorem, ensuring that the energy changes from dissipation and random force cancel, requires that wD (rij ) = [wR (rij )]2 ,

(11.8)

and that the noise intensity must be related to the friction coeﬃcient γ and the temperature T , just as is the case for Langevin dynamics: σ 2 = 2γkB T.

(11.9)

The form of one of the weight function is arbitrary; Groot and Warren (1997) chose for wR the same functional form as for the conservative force: r ij rij < 1, rij = 0 rij ≥ 1.

wR (rij )C = (1 − rij )

(11.10) (11.11)

11.2 Prescribing ﬂuid parameters The unit of length has been set by the choice of the maximum interaction range. The number density n (per cubic length unit) can be chosen, and, together with a choice of a, the strength of the conservative force, the pressure is ﬁxed. The pressure is found from the usual virial equation: p=

N 1 kB T + r ij · F C ij . V 3V

(11.12)

i,j>i

In the virial part also the total force may be used, but in equilibrium the contributions of friction and random force cancel. One may wish to match the isothermal compressibility with a desired value for a given liquid. This

11.3 Numerical solutions

is best expressed as the dimensionless value ∂p 1 −1 , κ = kB T ∂n

309

(11.13)

which has a value of 16 for water at 300 K and 30 for 1-propanol at 300 K. From a series of simulations Groot and Warren (1997) found that an p = nkB T + α , α = 0.101 ± 0.001. (11.14) kB T This determines an/kB T , which is equal to 75 for water. 11.3 Numerical solutions Since the force F(t) depends on the equal-time velocity v(t), the normal Verlet-type algorithms cannot be used, because they have the equal-time velocities v(t) available after a step that has already used the force F(t). This applies also to the velocity-Verlet version. If earlier velocities are used, the order of the algorithm degrades and the performance becomes unacceptable. Possibilities are to predict the velocity for the force calculation and correct it afterwards, or solve the velocity iteratively, requiring more than one force evaluation per step. Lowe (1999) has devised an alternative algorithm which adds a thermostat much like Andersen’s velocity rescaling. The equations of motion are integrated with the velocity-Verlet scheme, but in addition randomly selected pairs of particles exchange their relative velocity for a sample drawn from a Maxwellian distribution, in such a way that momentum (and angular momentum) is conserved. This solves problems with temperature drift that otherwise occur.

11.4 Applications Applications have been published in many diﬀerent ﬁelds, such as polymer rheology (Schlijper et al., 1995), rheology of colloidal suspensions (Koelman and Hoogerbrugge, 1993; Boek et al., 1997), ﬂow of DNA molecules in microchannels (Fan et al., 2003). The method can be applied to mixtures and to microphase separation (Groot and Warren, 1997). Figure 11.1 shows the impact and subsequent coalescence of two drops of liquid moving towards each other in a liquid environment in which the drops don’t mix. The simulation comprises 3.75 million DPD particles and was carried out by Florin O. Iancu, University of Delft, the Netherlands (Iancu, 2005). The collision is characterized by the dimensionless Weber number W e = ρDUr2 /Γ, where ρ is the density, D the diameter of the drops, Ur the relative velocity of

310

Dissipative particle dynamics

the drops just before impact and Γ the interfacial surface tension. At low Weber numbers the drops bounce oﬀ each other without mixing, and at higher Weber numbers they coalesce after impact. The DPD parameters in this case were a density of 10 particles per rc3 (rc being the cut-oﬀ range of the repulsive potential) and a repulsion parameter a (see (11.5)) of 14.5 mutually between drop particles or between environment particles, but 41.5 between drop particles and environment particles. This choice leads to an interfacial tension Γ = 28 (units of kB T /rc2 ). When the collision is oﬀ-center, the drops elongate before they coalesce and may even split up afterwards with formation of small satellite droplets.

11.4 Applications

311

Figure 11.1 Impact and coalescence of two liquid drops moving towards each other in an inmiscible liquid environment at Weber’s number of about 4, simulated with 3.75 million DPD particles (courtesy of Dr F.O. Iancu, University of Delft, the Netherlands)

Part II Physical and Theoretical Concepts

12 Fourier transforms

In this chapter we review the deﬁnitions and some properties of Fourier transforms. We ﬁrst treat one-dimensional non-periodic functions f (x) with Fourier transform F (k), the domain of both coordinates x and k being the set of real numbers, while the function values ∞ may be complex. The functions f and F are piecewise continuous and −∞ |f (x)| dx exists. The domain of x is usually called the real space while the domain of k is called reciprocal space. Such transforms are applicable to wave functions in quantum mechanics. In Section 12.6 we consider Fourier transforms for one-dimensional periodic functions, leading to discrete transforms, i.e., Fourier series instead of integrals. If the values in real space are also discrete, the computationally eﬃcient fast Fourier transform (FFT) results (Section 12.7). In Section 12.9 we consider the multidimensional periodic case, with special attention to triclinic periodic 3D unit cells in real space, for which Fourier transforms are useful when long-range forces are evaluated.

12.1 Deﬁnitions and properties The relations between f (x) and its Fourier transform (FT) F (k) are ∞ 1 F (k) exp(ikx) dk, (12.1) f (x) = √ 2π −∞ ∞ 1 f (x) exp(−ikx) dx. (12.2) F (k) = √ 2π −∞ √ The factors 1/ 2π are introduced for convenience in order to make the transforms symmetric; one could use any arbitrary factors with product 2π. The choice of sign in the exponentials is arbitrary and a matter of convention. Note that the second equation follows from the ﬁrst by using the deﬁnition 315

316

Fourier transforms

of the δ-function:

∞

exp[±ikx] dk = 2πδ(x),

(12.3)

−∞

and realizing that

∞

f (x) = −∞

δ(x − x) f (x ) dx .

(12.4)

The following relations are valid: (i) if f (x) is real then F (−k) = F ∗ (k) (ii) if F (k) is real then f (−x) = f ∗ (x); (iii) if f (x) is real and f (−x) = f (x) then F (k) is real and F (−k) = F (k) (cosine transform); (iv) if f (x) is real and f (−x) = −f (x) then F (k) is imaginary and F (−k) = −F (k) (sine transform); def

(v) the FT of g(x) = f (x + x0 ) is G(k) = F (k) exp(ikx0 ); def

(vi) the FT of g(x) = f (x) exp(ik0 x) is G(k) = F (k − k0 ); def

(vii) the FT of g(x) = df (x)/dx is G(k) = ikF (k); def

(viii) the FT of g(x) = xf (x) is G(k) = −i dF (k)/dk. 12.2 Convolution and autocorrelation The convolution h(x) of two functions f (x) and g(x) is deﬁned as ∞ def h(x) = f ∗ (ξ − x)g(ξ) dξ −∞ ∞ f ∗ (ξ)g(x + ξ) dξ, =

(12.5) (12.6)

−∞

with short notation h = f ∗ g. Its Fourier transform is √ H(k) = 2πF ∗ (k)G(k), and hence

∞

h(x) =

F ∗ (k)G(k) exp(ikx) dk.

(12.7)

(12.8)

−∞

√ def ∞ If h(x) = −∞ f (ξ − x)g(ξ) dξ then H(k) = 2πF (k)G(k). A special case is ∞ ∞ ∗ h(0) = f (x)g(x) dx = F ∗ (k)G(k) dk. −∞

−∞

(12.9)

12.3 Operators

317

The autocorrelation function is a self-convolution: ∞ def f ∗ (ξ − x)f (ξ) dξ, h(x) = −∞ √ H(k) = 2πF ∗ (k)F (k), ∞ F ∗ (k)F (k) exp(ikx) dk, h(x) = −∞ ∞ ∞ f ∗ (x)f (x) dx = F ∗ (k)F (k) dk. h(0) = −∞

(12.10) (12.11) (12.12) (12.13)

−∞

Equation (12.13)is known as Parseval’s theorem. ∞ ∗ It implies that, if the function f (x) is normalized in the sense that −∞ f f dx = 1, then its Fourier transform is, in the same sense in k-space, also normalized. We note that the deﬁnitions given here for square-integrable functions diﬀer from the autocorrelation and spectral density functions for inﬁnite time series discussed in Section 12.8 (page 325).

12.3 Operators When the function f ∗ f (x) is interpreted as a probability density, the expectation of some function of x (indicated by triangular brackets) is the average of that function over the probability density: ∞ def h(x)f ∗ f (x) dx. (12.14) h(x) = −∞

Functions of k are similarly deﬁned by averages over the probability density F ∗ F (k) in k-space: ∞ def h(k)F ∗ F (k) dk. (12.15) h(k) = −∞

It can be shown that for polynomials of k the average can also be obtained in x-space by ∞ ˆ (x) dx, h(k) = f ∗ (x)hf (12.16) −∞

ˆ is an operator acting on f (x) with the property that where h ˆ exp(ikx) = h(k) exp(ikx). h

(12.17)

ˆ = −i ∂ , h ∂x

(12.18)

Examples are h(k) = k,

318

Fourier transforms 2 ˆ =− ∂ , h ∂x2 n ˆ = i−n ∂ . h ∂xn

h(k) = k 2 , h(k) = k n ,

(12.19) (12.20)

Proof We prove (12.16). Insert the Fourier transforms into (12.16), using (12.3) and (12.17): ∞ ∞ ∞ ∞ 1 ∗ˆ ˆ ikx f hf dx = dx dk dk F ∗ (k )e−ik x F (k)he 2π −∞ −∞ −∞ −∞ ∞ ∞ ∞ 1 = dk dk F ∗ (k )F (k)h(k) dxei(k−k )x 2π −∞ −∞ ∞ −∞ = dkF ∗ (k)F (k)h(k) = h(k). −∞

In general, an operator Aˆ may be associated with a function A of x and/or k, and the expectation of A deﬁned as ∞ def ˆ (x) dx. A = f ∗ (x)Af (12.21) −∞

An operator Aˆ is hermitian if for any two quadratically integrable functions f (x) and g(x) ∞ ∗ ∞ ∞ ∗ ˆ ∗ ˆ f Ag dx = g Af dx = g Aˆ∗ f ∗ dx. (12.22) −∞

−∞

−∞

In particular this means that the expectation of a hermitian operator is real, as is immediately seen if we apply the hermitian condition to g = f . Operators that represent physical observables, meaning that expectations must be real physical quantities, are therefore required to be hermitian. It also follows that the eigenvalues of hermitian operators are real because the eigenvalue is the expectation of the operator over the corresponding eigenfunction (the reader should check this).

12.4 Uncertainty relations If we deﬁne the variances in x- and k-space as def

σx2 = (x − x)2 ,

(12.23)

def σk2 =

(12.24)

(k − k)2 ,

12.4 Uncertainty relations

319

we can prove that for any normalized function f (x) the product of the square root of these two variances (their standard deviations) is not less than one half: σx σk ≥ 12 .

(12.25)

This is the basis of the Heisenberg uncertainty relations for conjugate variables. Proof The proof1 starts with the Schwarz inequality for the scalar products of any two vectors u and v: (u, u)(v, v) ≥ (u, v)(v, u) = |(u, v)|2

(12.26)

which is valid with the deﬁnition of a scalar product of functions.2 ∞ def (u, v) = u∗ v dx. (12.27) −∞

This the reader can prove by observing that (u − cv, u − cv) ≥ 0 for any choice of the complex constant c, and then inserting c = (v, u)/(v, v). We make the following choices for u and v: u = (x − x)f (x) exp(ik0 x) d [f (x) exp(ik0 x)], v = dx

(12.28) (12.29)

where k0 is an arbitrary constant, to be determined later. The two terms on the left-hand side of (12.26) can be worked out as follows: ∞ (u, u) = (x − x)2 f ∗ f dx = σx2 (12.30) −∞

and

d ∗ d (v, v) = f exp(−ik0 x) f exp(ik0 x) dx dx −∞ dx 2

∞ d ∗ = − f exp(−ik0 x) f exp(ik0 x) dx dx2 −∞ ∞ ∞ f ∗ f dx − 2ik0 f ∗ f dx + k02 = −

∞

−∞

= k − 2k0 k + 2

1

2

−∞

k02 ,

(12.31)

See, e.g., Kyrala (1967). Gasiorowicz (2003) derives a more general form of the uncertainty relations, relating the product of the standard deviations of two observables A and B to their ˆ B]|. ˆ commutator: σA σb ≥ 12 |i[A, This deﬁnition applies to vectors in Hilbert space. See Chapter 14.

320

Fourier transforms

where the second line follows from the ﬁrst one by partial integration. Choosing for k0 the value for which the last form is a minimum: k0 = k, we obtain (v, v) = σk2 . Thus, (12.26) becomes σx2 σk2

≥ | ≥ 1 = 4 1 = 4

∞

u∗ v dx|2

−∞

∞

Re

(12.32)

−∞ ∞

−∞ ∞

2 u v dx ∗

∗

2

∗

(x − x)(f f + f f ) dx

d (x − x) (f ∗ f ) dx dx −∞ ∞ 2 1 1 = f ∗ f dx = . 4 4 −∞

2

(12.33)

Hence σx σk ≥ 12 . 12.5 Examples of functions and transforms In the following we choose three examples of real one-dimensional symmetric functions f (x) that represent a conﬁnement in real space with diﬀerent shape functions. All functions have expectations zero and are quadratically normalized, meaning that ∞ f 2 (x) dx = 1. (12.34) −∞

This implies that their Fourier transforms are also normalized and that the expectation of k is also zero. We shall look at the width of the functions in real and reciprocal space. 12.5.1 Square pulse The square pulse and its Fourier transform are given in Fig. 12.1. The equations are: 1 f (x) = √ , a = 0,

a 2 a |x| ≥ 2 |x| <

(12.35)

12.5 Examples of functions and transforms

321

2

0

−2

1

f(x)

1 −a/2 −1

0

a/2 0

F(k)

1

2

−4

−2

0

2

4

Figure 12.1 Square pulse f (x) with width a (x in units a/2, f in units a−1/2 ) and its transform F (k) (k in units 2π/a, F in units (a/2π)1/2 ). 1

1 F(k)

f(x)

0

−a

−2

−1

0

a 0

1

2

-3

-2

-1

0

1

2

3

Figure 12.2 Triangular pulse f (x) with width 2a (x in units a, f in units (2a/3)−1/2 ) and its transform F (k) (k in units 2π/a, F in units (3a/4π)1/2 ).

x = 0

(12.36)

1 σx2 = x2 = a2 12 & a sin φ , F (k) = 2π φ k = 0 σk2

(12.37) φ = 12 ka

= k = ∞ 2

σx σ k = ∞

(12.38) (12.39) (12.40) (12.41)

12.5.2 Triangular pulse The triangular pulse is a convolution of the square pulse with itself. See Fig. 12.2. The equations are: & 1 3 (a − |x|), |x| < a (12.42) f (x) = a 2a = 0, |x| ≥ a x = 0

(12.43)

322

Fourier transforms

1

1 f(x)

0

F(k)

0 −4

−2

0

2

4

−4

−2

0

2

4

Figure 12.3 Gaussian pulse f (x) with variance σ (x in units σ,√f in units √ (σ 2π)−1/2 and its transform F (k) (k in units 1/2σ, F in units (2σ/ 2π)−1/2 ).

1 σx2 = x2 = a2 10 & 3a sin2 φ F (k) = 4π φ2 k = 0 3 σk2 = k 2 = 2 a & 3 = 0.5477 σx σk = 10

(12.44) φ = 12 ka

(12.45) (12.46) (12.47) (12.48)

12.5.3 Gaussian function The Gaussian function (Fig. 12.3) is the Fourier transform of itself. the equations are: √ −1 x2 2 (12.49) f (x) = (σ 2π) exp − 2 4σ x = 0 (12.50) σx2 = x2 = σ 2 √ −1 k2 F (k) = (σk 2π) 2 exp − 2 ; 4σk k = 0 1 σk2 = k 2 = 2 4σ σx σk = 12

(12.51) 1 σk = 2σ

(12.52) (12.53) (12.54) (12.55)

So we see that – of the functions given here – the Gaussian wave function attains the smallest product of variances in real and reciprocal space. In

12.6 Discrete Fourier transforms

323

fact, the Gaussian function has the smallest possible product of variances of all (well-behaved) functions.

12.6 Discrete Fourier transforms Now consider periodic functions f (x) with periodicity a: f (x + na) = f (x), n ∈ Z.

(12.56)

The function is assumed to be piecewise continuous and absolutely integrable over the domain (0, a): a |f (x)| dx exists. (12.57) 0

The function f (x) can now be expressed in an inﬁnite series of exponential functions that are each periodic functions of x on (0, a): 2πinx f (x) = , (12.58) Fn exp a n∈Z

with 1 Fn = a

a

f (x) exp 0

−2πinx a

dx.

(12.59)

2πn , n ∈ Z, a

(12.60)

This can also be written as f (x) =

k

1 Fk = a

Fk eikx , k =

a

f (x)e−ikx dx.

(12.61)

0

a The validity can be checked by computing 0 [ k exp(ik x)] exp(−ikx) dx. All terms with k = k vanish, and the surviving term yields aFk . It is clear that f (x) is real when F−k = Fk∗ . In that case the transform can also be expressed in sine and cosine transforms: f (x) = 12 a0 + ∞ (ak cos kx + bk sin kx), k = 2πn ., a , n = 1, 2, . .(12.62) a k 1 ak = f (x) cos kx dx, (12.63) 2a 0 a 1 f (x) sin kx dx. (12.64) bk = 2a 0

324

Fourier transforms

12.7 Fast Fourier transforms A special form of discrete Fourier transforms is the application to periodic discrete functions known at regular intervals Δx = a/N , where N is the number of intervals within the period a. The function values are fn = f (xn ), with xn = na/N, n = 0, . . . , N − 1. The Fourier relations are now fn =

Fm =

N −1

Fm e2πinm/N ,

m=0 N −1

1 N

fn e−2πinm/N .

(12.65)

(12.66)

n=0

These relations are easily veriﬁed by inserting (12.65) into (12.66) and re alizing that n exp[2πi(m − m)n/N ] equals zero unless m = m. In the general case the arrays f and F are complex. One should view both arrays as periodic: they can be shifted to another origin if required. Consider the values fn as a variable in time t, deﬁned on a periodic interval [0, T ) and discretized in N small time intervals Δt. Thus T = N Δt. The data are transformed to a discretized frequency axis νm , with resolution Δν = 1/T and maximum frequency determined by νmax = 1/2Δt. The latter follows from Nyquist’s theorem stating that two data points per period of the highest frequency are suﬃcient and necessary to reproduce a function of time that is band-limited to νmax (i.e., which has no Fourier components beyond νmax ).3 The transform contains N frequency components between −νmax and νmax in steps of Δν. Because of the periodicity of Fm in [0, n), the negative frequencies are to be found for N/2 ≤ m < N : F−m = FN −m . In the special but common case that fn is real, the transform is ∗. described by a complex array of length N/2 because F−m = FN −m = Fm If a continuous function that contains frequencies ν > νmax is sampled at intervals Δt = 1/2νmax , these higher frequencies are aliased or folded back into the frequency domain [0, νmax ) and appear as frequencies 2νmax −ν, i.e., the signal is mirrored with respect to νmax . Such a “false” signal is called an alias. When noisy signals are sampled at intervals Δt, noise components with frequencies above 1/2Δt will appear as low-frequency noise and possible mask interesting events. In order to prevent aliasing, one should either apply a low-pass ﬁlter to the signal before sampling, or use oversampling with subsequent removal of the high-frequency components. The assumed periodicity of the series fn , n ∈ [0, N ), may also cause 3

If a series fn is interpolated with the sinc function: f (t) = n fn sin z/z, with z = π(t−ti )/Δt, then the Fourier transform of the resulting function vanishes for ν > 1/2Δt.

12.8 Autocorrelation and spectral density from FFT

325

artefacts when fn are samples over a period T of a truncated, but in principle inﬁnite series of data. In order to avoid unrealistic correlations between the selected time series and periodic images thereof, it is adviza ble to extend the data with a number of zeroes. This is called zero-padding and should ideally double the length of the array. The double length of the data series reﬁnes resolution of the frequency scale by a factor of two; of course the factual resolution of the frequency distribution is not changed just by adding zeroes, but the distributions look nicer. Distributions can be made smoother and wiggles in spectral lines, caused by sudden truncation of the data, can be avoided by multiplying the time data by a window function that goes smoothly to zero near the end of the data series. An example is given below. Since very fast algorithms (FFT, fast Fourier transform, invented by Cooley and Tukey, 1965)4 exist for this representation of Fourier transforms, the periodic-discrete form is most often used in numerical problems. A well-implemented FFT scales as N log N with the array size N . The most eﬃcient implementations are for values of N that decompose into products of small primes, such as powers of 2.

12.8 Autocorrelation and spectral density from FFT Consider an inﬁnite series of real data fn , n ∈ Z. Let the series be stationary in the sense that statistical properties (average, variance, correlation function) evaluated over a speciﬁc interval [i, . . . i + N ) are within statistical accuracy independent of the origin i. The autocorrelation function Ck , which is discrete in this case, is deﬁned as i+N −1 1 fn fn+k . N →∞ N

Ck = lim

(12.67)

n=i

The value of Ck does not depend on the origin i; C0 equals the mean square (also called the mean power per sample) of f . The function is symmetric: C−k = Ck . Generally the autocorrelation is understood to apply to a function with zero mean: when fn has zero mean, Ck tends to 0 for large k and C0 is the variance of f . If the mean of fn is not zero, the square of the mean is added to each Ck . The discrete autocorrelation function, as deﬁned above, can be viewed as a discretization of the continuous autocorrelation 4

See for a description Press et al. (1992) or Pang (1997). Python numarray includes an FFT module. A versatile and fast, highly recommended public-domain C subroutine FFT library is available from http://www.ﬀtw.org/ (“Fastest Fourier Transform in the West”).

326

Fourier transforms

function C(τ ) of a continuous stationary function of time f (t):5 1 C(τ ) = lim T →∞ T def

t0 +T

f (t)f (t + τ ) dt,

(12.68)

t0

where t0 is an arbitrary origin of the time axis. Note that C(0) is the mean power per unit time of f (t) and that C(−τ ) = C(τ ). The autocorrelation function can be estimated from a truncated series of data fn , n ∈ [0, N ) as Ck ≈ Cktrunc Cktrunc

=

trunc C−k

N −k−1 1 = fn fn+k (k ≥ 0), N −k

(12.69)

n=0

or from a periodic series of data fn , n ∈ Z, fn+N = fn , as Ck ≈ Ckper =

N −1 1 fn fn+k . N −k

(12.70)

n=0

The factor 1/(N − k) instead of 1/N in (12.70) corrects for the fact that k terms in the periodic sum have the value zero, provided the data series has been properly zero-padded. Without zero-padding it is better to use the factor 1/N ; now Ckper does not approximate Ck but a mixture of Ck and CN −k : N −1 k 1 N −k Ck + CN −k ≈ Ckper = fn fn+k . N N N

(12.71)

n=0

This makes the correlation function symmetric about N/2. The estimation is in all cases exact in the limit N → ∞, provided that the correlation dies out to negligible values above a given n. We now consider the Fourier transform of the continuous autocorrelation function. Because the autocorrelation function is real and even, its Fourier transform can be expressed as a cosine transform over positive time: ∞ S(ν) = 4 C(τ ) cos(2πντ ) dτ. (12.72) 0

S(ν) is a real even function of τ . The factor 4 is chosen such that the inverse 5

Of course, one may read any other variable, such as a spatial coordinate, for t.

12.8 Autocorrelation and spectral density from FFT

transform6 has a simple form:

327

∞

S(ν) cos(2πντ ) dν.

C(τ ) =

(12.73)

0

We see that the power per unit time C(0) equals the integral of S(ν) over all (positive) frequencies. Therefore we may call S(ν) the spectral density, as S(ν)dν represents the power density in the frequency interval dν. The spectral density can also be determined from the direct Fourier transform of the time function. Consider a time slice fT (t), t ∈ [0, T ), extended with zero for all other t, with its Fourier transform T f (t)e2πiνt dt, (12.74) FT (ν) = 0 ∞ f (t) = F (ν)e−2πiνt dt. (12.75) −∞

We can now see that the spectral density is also given by 1 ∗ FT (ν)FT (ν). T →∞ T

S(ν) = 2 lim

(12.76)

The right-hand side is the mean power (square of absolute value, irrespective of the phase) per unit of time expressed as density in the frequency domain. The factor 2 arises from the fact that S(ν) is deﬁned for positive ν while the right-hand side applies to the frequency domain (−∞, ∞). The equality of (12.72) and (12.76) is the Wiener–Khinchin theorem. Proof First we assume that the limit exists. It is intuitively clear that this is true for stationary time series that contain no constant or periodic terms.7 We skip further discussion on this point and refer the interested reader to the classic paper by Rice (1954) for details. Substituting C(τ ) from (12.68) into the deﬁnition of S(ν) given in (12.72), we ﬁnd ∞ 2 T S(ν) = lim dt dτ fT (t)fT (t + τ )e2πiντ . (12.77) T →∞ T 0 −∞ On the other hand, (12.76) can be written as T 2 T dt dt fT (t)e−2πiνt fT (t )e2πiνt S(ν) = lim T →∞ T 0 0 6

7

The deﬁnition of the transform diﬀers slightly from the deﬁnitions given in Section 12.1 on page 315, where 2πν is taken as reciprocal variable rather than ν. Because of this the factors √ 2π disappear. In fact, we do require that the integral from 0 to ∞ of the correlation function exists; this is not the case when constant or periodic components are present.

328

Fourier transforms

2 T →∞ T

T

= lim

T −t

dt 0

−t

dτ fT (t)fT (t + τ )e2πiντ .

(12.78)

Now, for large T , t “almost always” exceeds the time over which the correlation function is non-zero. Therefore the integral over τ “almost always” includes the full correlation function, so that the limits can be taken as ±∞. In the limit of T → ∞ this is exact. Combination of (12.73) and (12.76) shows that the autocorrelation function can be obtained from the inverse FT of the squared frequency amplitudes: 1 ∞ ∗ FT (ν)FT (ν) cos(2πντ ) dν C(τ ) = 2 lim T →∞ T 0 ∞ 1 FT∗ (ν)FT (ν)e−2πiντ dν. (12.79) = lim T →∞ T −∞ Note that the sign in the exponent is irrelevant because of the symmetry of F ∗F . The discrete periodic case is similar. Given a time series fn , periodic on [0, N ), we obtain the following exact relations: (i) Autocorrelation function Ckper deﬁned in (12.70). (ii) Fourier transform Fm of time series is deﬁned by (12.66). (iii) Spectral density from Fm : per = Sm

2 ∗ F Fm . N m

(12.80)

(iv) Autocorrelation function from inverse FFT of spectral density:

N/2−1

Ckper =

per Sm cos(2πmk/N ) =

m=0

N −1 1 per −2πimk/N Sm e . (12.81) 2N m=0

(v) Spectral density from FFT of autocorrelation function:

N/2−1 per Sm =4

k=0

Ckper cos(2πmk/N ) = 2

N −1

Ckper e2πimk/N .

(12.82)

k=0

When the data are samples of a continuous time series taken at intervals Δt during a total time span T = N Δt, the autocorrelation coeﬃcients Ckper provide an estimate for the continuous autocorrelation function C(kΔt) of (3.37). If zero-padding has been applied, the estimate is improved by scaling C(kΔt) with N/(N − k) (see (12.70) on page 326). Without zero-padding, Ckper provides an estimate for a mixture of C(kΔt) and C(T − kΔt), as

12.8 Autocorrelation and spectral density from FFT

329

given by (12.71). The spectral resolution is Δν = 1/T and the coeﬃcients Sm represent the “power” in a frequency range Δν. The continuous power density per unit of frequency S(ν), given by (12.72) and (12.76), equals Sm /Δν = T Sm . If the time series is longer than can be handled with FFT, one may break up the series into a number of shorter sections, determine the spectral density of each and average the spectral densities over all sections. One may then proceed with step (iv). This procedure has the advantage that a good error estimate can be made based on the variance of the set of spectra obtained from the sections. Spectra obtained from one data series will be noisy and – if the data length N greatly exceeds the correlation length of the data (as it should) – too ﬁnely grained. One may then apply a smoothing procedure to the spectral data. Beware that this causes a subjective alteration of the spectrum! This can be done by convolution with a local spread function, e.g., a Gaussian function,8 but the simplest way is to multiply the autocorrelation function with a window function which reduces the noisy tail of the correlation function. The smoothed spectral density is then recovered by FFT of the windowed autocorrelation function. The eﬀect is a convolution with the FT of the window function. If a Gaussian window is used, the smoothing is Gaussian as well. See the following example. In Fig. 12.4 an example is given of the determination of a smoothed spectral density from a given time series. The example concerns an MD simulation at 300 K of the copper-containing protein azurin in water and the question was asked which vibration frequencies are contained in the ﬂuctuation of the distance between the copper atom and the sulphur atom of a cysteine, one of the copper ligands. Such frequencies can be compared to experimental resonance-Raman spectra. A time slice of 20 ps with a resolution of 2 fs (10 000 data points) was considered (Fig. 12.4a). It was Fourier-transformed (Δν = 1012 /20 Hz = 50 GHz) and its power spectrum, which had no signiﬁcant components above a range of 400 points (20 THz), computed and plotted in Fig. 12.4b. The complete power spectrum was subsequently inversely Fourier transformed to the autocorrelation function (not shown), which was multiplied by a Gaussian window function with

8

See Press et al. (1992) for a discussion of optimal smoothing.

330

Fourier transforms

Cu-S distance (nm)

0.24 0.23 0.22 0.21 4

8

12

a

16

20 time (ps)

spectral intensity (a.u.) 8

spectral intensity (a.u.) 100

b

c

7

80

6

60

5 4

40

3 2

20

1 0 0

5 100

200

15 20 frequency (THz)

0

300 400 500 600 wavenumber (cm-1)

0

10

5 100

200

10 300

15 20 frequency (THz) 400 500 600 wavenumber (cm-1)

Figure 12.4 Fluctuating distance between Cu and S atoms in the copper protein azurin, from an MD simulation, and its spectral density. (a) Time series of 10 000 points, time step 2 fs, duration 20 ps. (b) Spectral intensity (square of absolute value of FFT) by direct FFT of time series. (c) The same after low-pass ﬁltering by applying a Gaussian window to the autocorrelation function (data from Marieke van de Bosch, Leiden).

s.d. of 300 points (600 fs). Fourier transformation then gave the smoothed spectrum of Fig. 12.4c. Here are the few Python lines that do the trick. python program 12.1 Spectrum from time series Computes the smoothed spectral density from a simulated time series. 01 02 03 04 05

from fftpack import fft,ifft # load data array f fdev=f-f.mean() N = len(fdev) F = fft(fdev)

12.9 Multidimensional Fourier transforms 06 07 08 09 10 11 12

331

FF = F.real**2 + F.imag**2 spec1 = FF[:400] acf = ifft(FF).real sigma = 300. window = exp(-0.5*(arange(N)/sigma)**2) acf2 = acf*window spec2 = fft(acf2).real[:400]

Comments Line 2: ﬁll array f with data read from ﬁle. Subtract average in line 4. Line 6 computes F ∗ F and line 7 produces the relevant part of the raw spectrum. Line 12 produces the relevant part of the smoothed spectrum. The window function is a Gaussian function with s.d. (σ) of 300 points. Only some 1000 points (3 σ) are relevant, but the full length of N (10 000) is retained to provide a dense grid for plotting.

12.9 Multidimensional Fourier transforms Fourier transforms are easily generalized to multidimensional periodic functions. For example, if f (x, y) is periodic in both x and y: f (x + n1 a, y + n2 b) = f (x, y), n1 , n2 ∈ Z2 , then (12.60) and (12.61) generalize to Fk1 k2 ei(k1 x+k2 y) , f (x, y) = k1

Fk1 k2

1 = ab

k2 a

dx

0

b

dy e−i(k1 x+k2 y) ,

(12.83)

(12.84) (12.85)

0

where k1 = 2πn1 /a and k2 = 2πn2 /b; n1 , n2 ∈ Z2 . In vector notation: (12.86) f (r) = Fk eik·r , k 1 Fk = dr f (r)e−ik·r , (12.87) V V where V is the volume ab . . .. Thus 3D FTs for periodic spatial functions with a rectangular unit cell, which fulﬁll the periodicity rule of (12.83), are simple products of three 1D FTs. This simple product decomposition does not apply to periodic spaces with monoclinic or triclinic unit cells.9 If the unit cell is spanned by (cartesian) base vectors a, b, c, the periodicity is expressed as f (r + n1 a + n2 b + n3 c) = f (r + Tn) = f (r), n ∈ Z3 , 9

See page 142 for the description of general periodicity in 3D space.

(12.88)

332

Fourier transforms

b* b a a*

Figure 12.5 A two-dimensional real lattice with base vectors a = (a, 0) and b = (0.25a, 0.5a).√The reciprocal vectors are a∗ = (1/a, −0.5/a) and b∗ = (0, 2/a). For a the value 2 is taken. Note that a large spacing in real space means a small spacing in “reciprocal space.”

where T is the transformation matrix ⎛ ⎞ ax bx cx T = ⎝ ay by cy ⎠ . az bz cz

(12.89)

This is not of the form of (12.83), but functions expressed in relative coordinates ρ = (ξ, η, ζ): r = ξa + ηb + ζc,

(12.90)

r = Tρ,

(12.91)

are periodic in the sense of (12.83): f (ρ) = f (ρ + n), n ∈ Z3 .

(12.92)

Fourier transforms now involve exp(±iκ · ρ), with κ = 2πm; m ∈ Z3 . These exponentials can be rewritten as exp(±ik · r) as follows (in matrix notation): κ · ρ = κT ρ = 2πmT T−1 r = kT r = k · r,

(12.93)

k = 2π(T−1 )T m.

(12.94)

if k is deﬁned as def

Deﬁning the (cartesian) reciprocal lattice vectors 10 a∗, b∗, c∗ by the rows of the inverse transformation matrix T−1 : ⎞ ⎛ ∗ ax a∗y a∗z (12.95) T−1 = ⎝ b∗x b∗y b∗z ⎠ , ∗ ∗ ∗ cx cy cz 10

Note that the asterisk does not represent a complex conjugate here.

Exercises

333

we see that k = 2π(m1 a∗ + m2 b∗ + m3 c∗ ).

(12.96)

With this deﬁnition of k in terms of reciprocal lattice vectors, the Fourier pair (12.86) and (12.87) remain fully valid. In crystallography, where f (r) represents an electron density, the quantities Fk are usually called structure factors and the indices m1 , m2 , m3 are often indicated by h, k, l. The volume V of the unit cell equals the determinant of T. The reciprocal lattice vectors have a scalar product of 1 with their corresponding base vectors and are perpendicular to the other base vectors: a · a∗ = 1,

(12.97)

a · b∗ = 0,

(12.98)

and similar for other products, as follows immediately from the deﬁnition of the reciprocal vectors. Fig. 12.5 shows a two-dimensional real lattice and the corresponding reciprocal lattice vectors.

Exercises 12.1 12.2

12.3 12.4 12.5 12.6 12.7 12.8

Show that hermitian operators have real eigenvalues. that the operator for k is hermitian. Use partial integration of Show f ∗ (∂g/∂x) dx; the product f ∗ g vanishes at the integration boundaries. Show that the expectation of k must vanish when the wave function is real. Show that the expectation value of k equals k0 if the wave function is a real function, multiplied by exp(ik0 ). Derive (12.62) to (12.64) from (12.60) and (12.61) and express ak and bk in terms of Fk . Show that ξ = a∗ · r, η = b∗ · r, ζ = c∗ · r. Prove (12.97) and (12.98). Derive the Fourier pair (12.72) and (12.73) from the original deﬁnitions in Section 12.1.

13 Electromagnetism

13.1 Maxwell’s equation for vacuum For convenience of the reader and for unity of notation we shall review the basic elements of electromagnetism, based on Maxwell’s equations. We shall use SI units throughout. Two unit-related constants ﬁgure in the equations: the electric and magnetic permittivities of vacuum ε0 and μ0 : 1 = 8.854 187 817 . . . × 10−12 F/m, (13.1) μ0 c2 μ0 = 4π × 10−7 N/A2 , (13.2) 1 (13.3) ε0 μ0 = 2 . c The basic law describes the Lorentz force F on a particle with charge q and velocity v in an electromagnetic ﬁeld: ε0 =

F = q(E + v × B).

(13.4)

Here, E is the electric ﬁeld and B the magnetic ﬁeld acting on the particle. The ﬁelds obey the four Maxwell equations which are continuum equations in vacuum space that describe the relations between the ﬁelds and their source terms ρ (charge density) and j (current density): div E = ρ/ε0 , div B = ∂B = curl E + ∂t 1 ∂E = curl B − 2 c ∂t Moving charges (with velocity v) produce j = ρv. 335

(13.5)

0,

(13.6)

0,

(13.7)

μ0 j.

(13.8)

currents: (13.9)

336

Electromagnetism

The charge density and current obey a conservation law, expressed as div j +

∂ρ = 0, ∂t

(13.10)

which results from the fact that charge ﬂowing out of a region goes at the expense of the charge density in that region.

13.2 Maxwell’s equation for polarizable matter In the presence of linearly polarizable matter with electric and magnetic susceptibilities χe and χm , an electric dipole density P and a magnetic dipole density (M ) are locally induced according to P = ε0 χe E, M = χm H.

(13.11) (13.12)

The charge and current densities now contain terms due to the polarization: ρ = ρ0 − div P , ∂P + curl M , j = j0 + ∂t

(13.13) (13.14)

where ρ0 and j 0 are the free or unbound sources. With the deﬁnitions of the dielectric displacement D and magnetic intensity 1 H and the material electric and magnetic permittivities ε and μ, D = ε0 E + P = εE, 1 1 B − M = B, H = μ0 μ

(13.15) (13.16)

the Maxwell equations for linearly polarizable matter are obtained:

1

div D = ρ0 ,

(13.17)

div B = 0, ∂B = 0, curl E + ∂t ∂D = j 0. curl H − ∂t

(13.18) (13.19) (13.20)

In older literature H is called the magnetic ﬁeld strength and B the magnetic induction.

13.3 Integrated form of Maxwell’s equations

337

The time derivative of D acts as a current density and is called the displacement current density. The permittivities are related to the susceptibilities: ε = (1 + χe )ε0 ,

(13.21)

μ = (1 + χm )μ0 .

(13.22)

ε is often called the dielectric constant, although it is advisable to reserve that term for the relative dielectric permittivity ε (13.23) εr = . ε0 13.3 Integrated form of Maxwell’s equations The Maxwell relations may be integrated for practical use. We then obtain: • the Gauss equation, relating the integral of the normal component of D over a closed surface to the total charge inside the enclosed volume: D · dS = q, (13.24) which leads immediately to the Coulomb ﬁeld of a point charge at the origin: q r D(r) = ; (13.25) 4π r3 • Faraday’s induction law, equating the voltage along a closed path with the time derivative of the total magnetic ﬂux through a surface bounded by the path: * ∂ Vind = E · dl = − B · dS; (13.26) ∂t • Ampere’s law, relating the magnetic ﬁeld along a closed path to the total current i through a surface bounded by the path: * ∂ D · dS. (13.27) H · dl = i+ ∂t 13.4 Potentials It is convenient to describe electromagnetic ﬁelds as spatial derivatives of a potential ﬁeld. This can only be done if four quantities are used to describe the potential, a scalar potential φ and a vector potential A: E = − grad φ − B = curl A.

∂A , ∂t

(13.28) (13.29)

338

Electromagnetism

The deﬁnitions are not unique: the physics does not change if we replace φ by φ−∂f /∂t and simultaneously A by A− grad f , where f is any diﬀerentiable function of space and time. This is called the gauge invariance. Therefore the divergence of A can be chosen at will. The Lorentz convention is div A +

1 ∂φ = 0, c2 ∂t

implying (in a vacuum) that 1 ∂2 ∇2 − 2 2 φ = −ρ0 /ε0 , c ∂t 1 ∂2 2 ∇ − 2 2 A = −μ0 j 0 . c ∂t

(13.30)

(13.31) (13.32)

13.5 Waves The Maxwell equations support waves with the velocity of light in vacuum, as can be seen immediately from (13.31) and (13.32). For example, a linearly polarized electromagnetic plane wave in the direction of a vector k, with wave length 2π/k and frequency ω/2π, has an electric ﬁeld E(r, t) = E 0 exp[i(k · r − ωt)],

(13.33)

where E 0 (in the polarization direction) must be perpendicular to k, and a magnetic ﬁeld 1 (13.34) B(r, t) = k × E(r, t). ω The wave velocity is ω = c. k

(13.35)

The wave can also be represented by a vector potential: A(r, t) =

i E(r, t). ω

(13.36)

The scalar potential φ is identically zero. Waves with A = 0 cannot exist. The vector Σ=E×H

(13.37)

is called the Poynting vector. It is directed along k, the direction in which the wave propagates, and its magnitude equals the energy ﬂux density, i.e., the energy transported by the wave per unit of area and per unit of time.

13.6 Energies

339

13.6 Energies Electromagnetic ﬁelds “contain” and “transport” energy. The electromagnetic energy density W of a ﬁeld is given by W = 12 D · E + 12 B · H.

(13.38)

In vacuum or in a linearly polarizable medium with time-independent permittivities, for which D = εE and B = μH, the time dependence of W is dW = − div Σ − j · E. (13.39) dt Proof Using the time-independence of ε and μ, we can write ˙ =E·D ˙ +H ·B ˙ W (to prove this in the case of tensorial permittivities, use must be made of ˙ by the fact that ε and μ are symmetric tensors). Now we can replace D ˙ curl H −j (see (13.20)) and B by − curl E (see (13.19)) and use the general vector equality div (A × B) = B · curl A − A · curl B.

(13.40)

Equation (13.39) follows. Equation (13.39) is an energy-balance equation: − div Σ is the energy ﬂowing out per unit volume due to electromagnetic radiation; −j · E is the energy taken out of the ﬁeld by the friction of moving charges and dissipated into heat (the Joule heat). Note that this equation is not valid if permittivities are time-dependent. For quasi-stationary ﬁelds, where radiation does not occur, the total ﬁeld energy of a system of interacting charges and currents can also be expressed in terms of the source densities and potentials as follows: 1 Uﬁeld = W (r) dr = (13.41) 2 (ρφ + j · A) dr, where integration of W is over all space, while the sources ρ and j are conﬁned to a bounded volume in space. Proof Quasi-stationarity means that the following equations are valid: E = − grad φ, div D = ρ, B = curl A,

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Electromagnetism

curl H = j. Using the ﬁrst two equations we see that D · E dr = − D · grad φ dr = φ div D dr = ρφ dr. The third integral follows from the second by partial integration, whereby the integral over the (inﬁnite) boundary vanishes if the sources are conﬁned to a bounded volume. Similarly B · H dr = − ( curl A) · H dr = A · curl H dr = j · A dr. The reader is invited to check the partial integration result by writing the integrand ount in all coordinates. This equation is often more convenient for computing energies than the ﬁeld expression (13.38). Both expressions contain a self-energy for isolated charges that becomes singular for delta-function point charges. This self-energy is not taken into account if the Coulomb interaction energy between a system of point charges is considered. Thus for a set of charges qi at positions r i , the total interaction energy is (13.42) Uint = 12 i qi (φi − φself i ), where = φi − φself i

qj 1 . 4πε |r j − r i |

(13.43)

j =i

This is indeed the usual sum of Coulomb interactions over all pairs: 1 qi qj . (13.44) Uint = 4πε |r j − r i | i<j

The factor

1 2

in (13.42) compensates for the double counting of all pairs.

13.7 Quasi-stationary electrostatics In the vast majority of molecular systems of interest, motions are slow enough that the time dependence in the Maxwell equations can be neglected, and the magnetic eﬀects of net currents are also negligible. The relations now simplify considerably. There are no magnetic ﬁelds and waves are no longer supported. The result is called electrostatics, although slow time dependence is not excluded.

13.7 Quasi-stationary electrostatics

341

13.7.1 The Poisson and Poisson–Boltzmann equations Since the curl of the electric ﬁeld E is now zero, E can be written as a pure gradient of a potential E = − grad φ.

(13.45)

In a linear dielectric medium, where the dielectric constant may still depend on position, we obtain from (13.15) and (13.17) the Poisson equation div (ε grad φ) = −ρ,

(13.46)

which simpliﬁes in a homogeneous dielectric medium to ρ ∇2 φ = − . ε

(13.47)

Here, ρ is the density of free charges, not including the “bound” charges due to divergence of the polarization. We drop the index 0 in ρ0 , as was used previously. In ionic solutions, there is a relation between the charge density and the potential. Assume that at a large distance from any source terms, where the potential is zero, the electrolyte contains bulk concentrations c0i of ionic species i, which have a charge (including sign) of zi e per ion.2 Electroneutrality prescribes that c0i zi = 0. (13.48) i

In a mean-ﬁeld approach we may assume that the concentration ci (r) of each species is given by its Boltzmann factor in the potential φ(r): zi eφ 0 ci (r) = ci exp − , (13.49) kTB so that, with ρ=F

ci z i ,

(13.50)

i

where F is the Faraday constant (96 485.338 C), we obtain the Poisson– Boltzmann equation for the potential: zi F φ 0 div (ε grad φ) = −F . (13.51) ci zi exp − RT i

In the Debye–H¨ uckel approximation the exponential is expanded and only 2

Note that concentrations must be expressed in mol/m3 , if SI units are used.

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Electromagnetism

the ﬁrst two terms are kept.3 Since the ﬁrst term cancels as a result of the electroneutrality condition (13.48), the resulting linearized Poisson– Boltzmann equation is obtained: div (ε grad φ) =

F2

0 2 i ci z i

φ. (13.52) RT The behavior depends only on the ionic strength of the solution, which is deﬁned as (13.53) I = 12 i c0i zi2 . Note that for a 1:1 electrolyte the ionic strength equals the concentration. In dielectrically homogeneous media the linearized Poisson–Boltzmann equation gets the simple form ∇2 φ = κ2 φ,

(13.54)

where κ is deﬁned by 2IF 2 . (13.55) εRT The inverse of κ is the Debye length, which is the characteristic distance over which a potential decays in an electrolyte solution. The Poisson equation (13.46) or linearized Poisson–Boltzmann equation (13.52) can be numerically solved on a 3D grid by iteration.4 In periodic systems Fourier methods can be applied and for arbitrarily shaped systems embedded in a homogeneous environment, boundary element methods are available.5 The latter replace the inﬂuence of the environment by a boundary layer of charges or dipoles that produce exactly the reaction ﬁeld due to the environment. For simple geometries analytical solutions are often possible, and in Sections 13.7.2, 13.7.3 and 13.7.4 we give as examples the reaction potentials and ﬁelds for a charge, a dipole and a charge distribution in a sphere, embedded in a polarizable medium. Section 13.7.5 describes the generalized Born approximation for a system of embedded charges. The following boundary conditions apply at boundaries where the dielectric properties show a stepwise change. Consider a planar boundary in the x, y-plane at z = 0, with ε = ε1 for z < 0 and ε = ε2 for z > 0, without free κ2 =

3

4 5

For dilute solutions the Debye–H¨ uckel approximation is appropriate. In cases where it is not appropriate, the full Poisson–Boltzmann equation is also not adequate, since the mean-ﬁeld approximation also breaks down. Interactions with individual ions and solvent molecules are then required. In general, when κ−1 (see (13.55)) approaches the size of atoms, the mean-ﬁeld approximation will break down. A popular program to solve the PB equation on a grid is DELPHI, see Nicholls and Honig (1991). Juﬀer et al. (1991).

13.7 Quasi-stationary electrostatics 2 ε2

φ2 (0)

1 ε1

φ1 (0)

E2x E1x

z 6

343

6 6 6 6 D2z 6 6 6 6

D1z

Figure 13.1 Boundary conditions on a planar discontinuity: potential φ, tangential electric ﬁeld Ex and perpendicular displacement Dz are continuous.

charge density at the surface (Fig. 13.1). Since the potential is continuous over the surface, its derivative along the surface, i.e., the component of the electric ﬁeld along the surface, is also continuous: Ex (z ↑ 0) = Ex (z ↓ 0), Ey (z ↑ 0) = Ey (z ↓ 0).

(13.56)

Since there is no free charge at the boundary, there is no source for D, and hence div D = 0. This implies that the ingoing ﬂux of D through one plane of a cylindrical box equals the outgoing ﬂux through the other plane at the other side of the boundary. Hence Dz (z ↑ 0) = Dz (z ↓ 0).

(13.57)

13.7.2 Charge in a medium Our ﬁrst example is a charge q in a cavity (ε0 ) of radius a in a homogeneous dielectric environment with ε = εr ε0 . According to (13.25), the dielectric displacement at a distance r > a equals q r . (13.58) D(r) = 4πr2 r The ﬁeld energy outside the cavity is ∞ 1 q2 Uﬁeld = . (13.59) D2 (r)4πr2 dr = 2ε a 8πεa The self-energy outside the cavity is the same with ε = ε0 and the excess ﬁeld energy due to the polarization of the environment, which is negative, is 1 q2 UBorn = − . (13.60) 1− 8πε0 a εr This polarization energy is often called the dielectric solvation energy or Born energy after Born (1920), who ﬁrst described this term. It is the (free)

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Electromagnetism

energy change when a charge q is moved from vacuum into a cavity with radius a in a dielectric continuum. Another description of the Born energy is by using the alternative ﬁeld energy description of (13.41) in terms of charge-potential products. The polarizable environment produces a potential φRP at the position of the charge, called the reaction potential, and the energy in that potential is UBorn = 12 qφRP .

(13.61)

The reaction potential is the summed potential of all induced dipoles in the medium: ∞ 1 P (r) 4πr2 dr, (13.62) φRP = − 4πε0 a r2 where

1 q 1 1 D= . P = 1− 1− εr 4π εr r 2

Integration yields φRP

q =− 4πε0 a

1 . 1− εr

(13.63)

(13.64)

This, of course, yields the same Born energy as was derived directly from integrating the ﬁeld energy in (13.60). How does the solvation energy of a charge in a cavity behave in the case that the medium is an electrolyte solution? Let us compute the reaction potential, i.e., the excess potential at r = 0 due to the presence of the medium beyond r = a. As before, the dielectric displacement is continuous at r = a, and therefore q D(a) = (13.65) 4πa2 is also valid at the boundary in the medium, so that the boundary conditions for φ are dφ q =− , φ(r) → 0 for r → ∞. (13.66) dr a 4πεa2 The diﬀerential equation for φ in the range r ≥ a is F 0 zi F φ 2 ci zi exp − ∇ φ=− ε RT

(13.67)

i

This equation can be solved numerically, given the ionic composition of

13.7 Quasi-stationary electrostatics

345

the medium. In the Debye–H¨ uckel approximation, and using the radial expression for the ∇2 operator, the equation simpliﬁes to 1 d 2 dφ r = κ2 φ, (13.68) r2 dr dr with the boundary conditions given above. The solution, valid for r ≥ a, is φ(r) =

q exp[−κ(r − a)] . 4πε(1 + κa)r

(13.69)

The potential in the cavity (r ≤ a) is given by D(r) = −ε0

q dφ = , dr 4πr2

(13.70)

and hence φ(r) =

q + φRP , 4πε0 r

(13.71)

where φRP is a constant given by the boundary condition that φ is continuous at r = a. This constant is also the excess potential (the reaction potential) due to the mean-ﬁeld response of the medium. Applying that boundary condition we ﬁnd 1 q φRP = − 1− . (13.72) 4πε0 a εr (1 + κa) The excess energy of the charge – due to interaction with the medium outside the cavity in excess to the vacuum self-energy – is, as in the previous case, given by Uexc = 12 qφRP .

(13.73)

We see that for zero ionic strength the Born reaction potential (13.60) is recovered, but that for large ionic strength the screening is more eﬀective, as if the dielectric constant of the medium increases. The excess energy due to κ: 1 q2 κ , (13.74) Uexc (κ) − Uexc (κ = 0) = − 2 4πε0 εr (1 + κa) is the (free) energy resulting from transferring the cavity with charge from an inﬁnitely dilute solution to the electrolyte (assuming the dielectric constant does not change). This term is due to the mean-ﬁeld distribution of ions around a charge: the counterions are closer and produce a negative energy. This term is responsible for the reduction of ionic chemical potentials in electrolyte solutions, proportional to the square root of the ionic concentration.

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Electromagnetism

13.7.3 Dipole in a medium The next example is the dielectric solvation energy for a dipole in the center of a spherical cavity with radius a in a medium with homogeneous dielectric constant ε. The problem is similar as the previous case of a charge, except that we cannot make use of spherical symmetry. Let us choose the z-axis in the direction of the dipole moment μ situated at r = 0, and use spherical coordinates r, θ (polar angle with respect to the z-axis), and φ (azimuthal angle of rotation around z, which must drop out of the problem for reasons of symmetry). For regions of space where there are no sources, i.e., for our problem everywhere except for r = 0, the Poisson equation (13.47) reduces to the Laplace equation ∇2 φ = 0,

(13.75)

which has the general solution in spherical coordinates φ(r, θ, φ) =

∞ l

l m −l−1 (Am )Plm (cos θ) exp(imφ), l r + Bl r

(13.76)

l=0 m=−l

where Plm are Legendre functions and A and B are constants that must follow from boundary conditions. In a bounded region of space (e.g., r ≤ a), the rl solutions are acceptable, but in an unbounded region with the requirement that φ → 0 for r → ∞, only the r−l−1 solutions are acceptable. The r−l−1 solutions are only acceptable for r → 0 if there is a singularity due to a source at r = 0. The singularity determines which angular term is acceptable. In the case of a dipole source, the singularity has a cosine dependence on θ (l = 1), and only the l = 1 terms need be retained. From these considerations, we can write the potentials in the cavity and in the medium as μ cos θ + br cos θ 4πε0 r2 c cos θ φmed (r) = (r ≥ a), r2 φcav (r) =

(r ≤ a),

(13.77) (13.78)

where b and c are constants to be determined from the boundary conditions at r = a: (i) φ is continuous: φcav (a) = φmed (a);

(13.79)

13.7 Quasi-stationary electrostatics

(ii) D is continuous in the radial direction: med cav dφ dφ =ε . ε0 dr dr a a

347

(13.80)

Applying these conditions gives straightforwardly b=−

μ 2(εr − 1) . 4πε0 a3 2εr + 1

(13.81)

The term br cos θ = bz in the cavity potential is simply the potential of a homogeneous electric ﬁeld in the z-direction. This is the reaction ﬁeld resulting from the polarization in the medium, in the direction of the dipole, and with magnitude −b: ERF =

μ 2(εr − 1) 4πε0 a3 2εr + 1

(13.82)

The energy of the dipole in the reaction ﬁeld is 1 μ2 2(εr − 1) . URF = − μERF = − 2 8πε0 a3 2εr + 1 We can now specify the potential anywhere in space: μ cos θ 1 r 2(εr − 1) cav (r ≤ a), − φ (r) = 4πε0 r2 a3 2εr + 1 μ cos θ 3 (r ≥ a). φmed (r) = 2 r 2εr + 1

(13.83)

(13.84) (13.85)

In the presence of ions in the medium, a similar reasoning can be followed as we used in deriving (13.72) for a single charge. The result is ERF = with

ε = εr

μ 2(ε − 1) , 4πε0 a3 2ε + 1

κ2 a2 1+ 2(1 + κa)

(13.86)

.

(13.87)

The eﬀect of the ionic strength is to increase the eﬀective dielectric constant and therefore the screening of the dipolar ﬁeld, just as was the case for the reaction potential of a charge. But the extra screening has a diﬀerent dependence on κ. Let us ﬁnally consider – both for a charge and for a dipole – the special case that either ε = ∞ or κ = ∞ (these apply to a conducting medium). In

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Electromagnetism

this case q , 4πε0 a μ = . 4πε0 a3

φRP = −

(13.88)

E RF

(13.89)

The potential and the ﬁeld of the source now vanish at the cavity boundary and are localized in the cavity itself.

13.7.4 Charge distribution in a medium Finally, we consider a charge distribution in a spherical cavity (ε0 ) of radius a, centered at the origin of the coordinate system, embedded in a homogeneous dielectric environment with ε = εr ε0 and possibly with an inverse Debye length κ. There are charges qi at positions r i within the cavity (ri < a). There are two questions to be answered: (i) what is the reaction potential, ﬁeld, ﬁeld gradient, etc., in the center of the cavity? This question arises if Coulomb interactions are truncated beyond a cut-oﬀ radius rc and one wishes to correct for the inﬂuence of the environment beyond rc (see Section 6.3.5 on page 164); (ii) what is the reaction ﬁeld due to the environment at any position within the sphere? This question arises if we wish to ﬁnd energies and forces of a system of particles located within a sphere but embedded in a dielectric environment, such as in molecular dynamics with continuum boundary conditions (see Section 6.2.2 on page 148). The ﬁrst question is simple to answer. As is explained in the next section (Section 13.8), the potential outside a localized system of charges can be described by the sum of the potentials of multipoles, each localized at the center of the system of charges, i.e., at the center of the coordinate system. The simplest multipole is the monopole Q = i qi ; it produces a reaction potential ΦRP at the center given by (13.72), replacing the charge in the center by the monopole charge. The reaction potential of the monopole is homogeneous and there are no reaction ﬁelds. The next multipole term is the dipole μ = i qi r i , which leads to a reaction ﬁeld ERF at the center given by (13.86) and (13.87). The reaction ﬁeld is homogeneous and there is no reaction ﬁeld gradient. Similarly, the quadrupole moment of the distribution will produce a ﬁeld gradient at the center, etc. When the system is described by charges only, one needs the ﬁelds to compute forces, but ﬁeld gradients are not needed and higher multipoles than dipoles are not

13.7 Quasi-stationary electrostatics

s qim = − (εr −1) a q (εr +1) s

62 a /s 6 a

349

θ sq

s6

r ε0

ε = εr ε 0

Figure 13.2 A source charge q in a vacuum sphere, embedded in a dielectric medium, produces a reaction potential that is well approximated by the potential of an image charge qim situated outside the sphere.

required. However, when the system description involves dipoles, one needs the ﬁeld gradient to compute forces on the dipole and the reaction ﬁeld gradient of the quadrupole moment of the distribution would be required. The second question is considerably more complicated. In fact, if the system geometry is not spherical, full numerical Poisson (or linearized PoissonBoltzmann) solutions must be obtained, either with ﬁnite-diﬀerence methods on a grid or with boundary-element methods on a triangulated surface. For a charge distribution in a sphere (ε0 ), embedded in a homogeneous dielectric environment (ε = εr ε0 ), Friedman (1975) has shown that the reaction ﬁeld is quite accurately approximated by the Coulomb ﬁeld of image charges outside the sphere. The approximation is good for εr 1. We shall not repeat the complete derivation (which is based on the expansion of the potential of an excentric charge in Legendre polynomials) and only give the results. Consider a charge q, positioned on the z-axis at a distance s from the center of a sphere with radius a (see Fig. 13.2). The potential inside the sphere at a point with spherical coordinates (r, θ) is given by the direct Coulomb potential of the charge plus the reaction potential ΦR (r, θ): rs n q n+1 (1 − εr ) Pn (cos θ). 4πε0 a n + εr (n + 1) a2 ∞

ΦR (r, θ) =

(13.90)

n=0

Here Pn (cos θ) are the Legendre polynomials (see page 361). Because of the axial symmetry of the problem, the result does not depend on the azimuthal angle φ of the observation point. This equation is exact, but is hard to

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Electromagnetism

evaluate. Now, by expanding the ﬁrst term in the sum: n+1 1 = (1 + c + c2 + · · ·), n + εr (n + 1) εr + 1

(13.91)

where c=

1 , (εr + 1)(n + 1)

(13.92)

one obtains an expansion of ΦR (r, θ): (0)

(1)

(2)

ΦR = ΦR + ΦR + ΦR + · · · .

(13.93)

The exact expression for the terms in this expansion is n ∞ r q (εr − 1) a (k) k ΦR (r, θ) = − (n + 1) Pn (cos θ). (εr + 1)k+1 s 4πε0 (a2 /s) a2 /s n=0 (13.94) (0) The zero-order term ΦR is just the Legendre-polynomial expansion (for r < a2 /s) of the potential at (r, θ) due to a charge qim at a position a2 /s on the z-axis:

2 −1/2 a2 qim a2 (0) 2 + r − 2r cos θ ΦR (r, cos θ) = 4πε0 s s qim , (13.95) = 4πε0 |r − r im | with qim = − (1)

(εr − 1) a q. (εr + 1) s

(13.96) (0)

The ﬁrst-order term ΦR is considerably smaller than ΦR /(εr + 1), which (0) is also true for the ratio of subsequent terms, so that ΦR is a good approximation to the exact reaction potential when εr 1. So, in conclusion: the reaction potential of a source charge q at position (s, 0) within a sphere (vacuum, ε0 ) of radius a is well approximated by the Coulomb potential (in vacuum) of an image charge qim (13.96) located at position r im = (a2 /s, 0) outside the sphere on the same axis as the source. The interaction free energy of the source q at distance s from the center (s < a) with its own reaction potential in the image approximation is q2 a2 1 (0) 1 εr − 1 UR (s) = qΦR (s) = − . (13.97) 2 2 εr + 1 4πε0 a a2 − s2

13.7 Quasi-stationary electrostatics

351

We can compare this result with the exact Born free energy of solvation. For a charge in the center (s = 0), we obtain q2 1 εr − 1 UR (0) = − , (13.98) 2 εr + 1 4πε0 a while the Born energy, according to (13.60), equals q2 1 εr − 1 UBorn = − . 2 εr 4πε0 a

(13.99)

The diﬀerence is due to the neglect of higher order contributions to the reaction potential and is negligible for large εr . We can also compare (13.97) with the exact result in the other limit: s → a (d a) approaches the case of a charge q at a distance d = a − s from a planar surface. The source is on the vacuum side; on the other side is a medium (εr ). The exact result for the reaction potential is the potential of an image charge qim = −q(εr − 1)/(εr + 1) at the mirror position at a distance d from the plane in the medium (see Exercise 13.3). In this limit, (13.97) simpliﬁes to 1 q2 1 εr − 1 UR (d) = − , (13.100) 2 εr + 1 4πε0 2d which is exactly the free energy of the charge in the reaction potential of its mirror image. When the sphere contains many charges, the reaction potentials of all charges add up in a linear fashion. Thus all charges interact not only with all other charges, but also with the reaction potential of itself (i.e., with its own image) and with the reaction potentials (i.e., the images) of all other charges.

13.7.5 The generalized Born solvation model When charges are embedded in an irregular environment, e.g. in a macromolecule that itself is solvated in a polar solvent, the electrical contribution to the free energy of solvation (i.e., the interaction energy of the charges with the reaction potential due to the polar environment) is hard to compute. The “standard” approach requires a time-consuming numerical solution of the Poisson (or linearized Poisson–Boltzmann) equation. In simulations with implicit solvent, the extra computational eﬀort destroys the advantage of omitting the solvent and explicit solvent representation is often preferred. Therefore, there is a pressing need for approximate solutions that can be rapidly evaluated, for simulations of (macro)molecules in an implicit solvent

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Electromagnetism

(see Section 7.7 on page 234). The generalized Born solvation model was invented to do just that. The original introduction by Still et al. (1990) was followed by several reﬁnements and adaptations,6 especially for application to macromolecules and proteins (Onufriev et al., 2004). The general problem is how to compute forces and energies for simulations of explicit (macro)molecules in an implicit polar solvent. The direct interactions between the explicit particles in the macromolecule (described by the “vacuum energy” Vvac ) must be augmented by the solvation free energy ΔGsolv between these particles and the solvent. The total solvation free energy consists of an electrostatic term ΔGel and a surface term ΔGsurf . Consider the following sequence of processes: (i) (ii) (iii) (iv)

start with the system in vacuum; remove all charges, i.e., remove the direct Coulomb interactions; solvate the uncharged system, i.e., add the surface free energy; add all charges back in the presence of the solvent.

The total potential of mean force is mf Vtot = Vvac + ΔGsurf + ΔGel .

(13.101)

The surface free energy is usually taken to be proportional to the solvent accessible surface area, with a proportionality constant derived from the experimental free energy of solvation of small molecules. Onufriev et al. (2004) quote a value of 0.005 kcal/mol par ˚ A2 = 2.1 kJ/mol per nm2 , but this value may be diﬀerentiated depending on the particular atom type. The free energy of adding all charges back in the presence of the solvent is the total electrostatic interaction in the polarizable medium. From this the direct Coulomb interaction (in vacuum) should be subtracted in order to obtain ΔGel , because the direct Coulomb interaction has been removed in step (ii) above. Now consider the total electrostatic interaction for a dilute system of charges qi , each in a (small) spherical vacuum cavity with radius Ri centered at position r i , embedded in a medium with relative dielectric constant εr (“dilute liquid of charges-in-cavities”). When all distances r ij are large compared to all radii Ri , the total electrostatic energy is qi qj 1 qi2 1 1− − . (13.102) 4πε0 Uel = εr rij 2 εr Ri i<j

6

i

To mention a few modiﬁcations of the expression (13.105) to compute the eﬀective “distance” GB : Hawkins et al. (1996) add another parameter to the term in square brackets; Onufriev et fij GB )/ε ) to take care of an ionic strength in the solvent; al. (2004) modify the 1/εr to exp(−κfij r Schaefer and Karplus (1996) retain a dielectric constant in the macromolecule.

13.8 Multipole expansion

353

Here, the ﬁrst term is the electrostatic energy of a distribution of charges in a medium (see (13.44) on page 340) and the second term is the sum of the Born energies of the charges (see (13.60) on page 343). After subtracting the direct vacuum Coulomb energy we obtain 1 qi qj 1 1− , (13.103) 4πε0 ΔGel = − 2 εr fijGB i j with fijGB = rij

for i = j,

= Ri

for i = j.

(13.104)

Still et al. (1990) propose the following form for fijGB that includes the case of the dilute liquid of charges-in-cavities, but has a much larger range of validity:

1/2 2 r ij 2 + Ri Rj exp − . (13.105) fijGB = rij 4Ri Rj The “eﬀective Born radii” Ri are to be treated as parameters that depend on the shape of the explicitly treated system and the positions of the charges therein. They are determined by comparison with Poisson–Boltzmann calculations on a grid, by free energy perturbation (or integration) calculations of charging a molecule in explicit solvent or from experimental solvation free energies. The eﬀective Born radius is related to the distance to the surface as we can see from the following example. Consider a charge q situated in a spherical cavity at a distance d from the surface. This is equivalent to the situation treated in Section 13.7.4, see Fig. 13.2, with s = a − d. The solvation free energy in the image approximation (13.97) then becomes 4πε0 UR = −

q2 1 (εr − 1) , 2 (εr + 1) 2d(1 − d/2a)

(13.106)

which is nearly equivalent to the GB equation with an eﬀective Born radius of 2d(1−d/2a). This equals twice the distance to the surface of the sphere for small distances, reducing to once the distance when the charge approaches the center. 13.8 Multipole expansion Consider two groups of charges, A and B, with qi at r i , i ∈ A and qj at r j , j ∈ B (see Fig. 13.3). Each group has a deﬁned central coordinate r A

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Electromagnetism

qi r i rA qj r j rB

A B

Figure 13.3 Two interacting non-overlapping groups of charges

and r B , and the groups are non-overlapping. The groups may, for example, represent diﬀerent atoms or molecules. For the time being the medium is taken as vacuum (ε = ε0 ) and the charges are ﬁxed; i.e., there is no polarizability. It is our purpose to treat the interaction energy of the two distributions in terms of a multipole expansion. But ﬁrst we must clearly deﬁne what the interaction energy is. The total energy of the two groups of charges is, according to (13.42), and omitting the self-energy terms in the potential: (13.107) U = 12 i∈A qi φ(r i ) + 12 j∈B qi φ(r j ). Here φ(r i ) is the sum of φA (r i ) due to all other charges in A, and φB (r i ) due to all charges in B. Furthermore, UA = 12 i∈A qi φA (r i ) is the internal electrostatic energy of group A (and, similarly, UB ). The total electrostatic energy equals U = UA + UB + UAB ,

(13.108)

with UAB being the interaction energy between group A and group B: (13.109) UAB = 12 i∈A qi φB (r i ) + 12 j∈B qj φA (r j ). Both terms in UAB are equal, which can be easily veriﬁed by inserting 1 qj , i∈A φB (r i ) = 4πε0 rij j∈B

and φA (r j ) =

1 qi , i∈B 4πε0 rij i∈A

13.8 Multipole expansion

355

into (13.109). Therefore the interaction energy UAB can also be written as qi φB (r i ), (13.110) UAB = i∈A

with φB (r) =

qj 1 . 4πε0 |r − r j |

(13.111)

j∈B

Thus we consider B as the source for the potential acting on A. By omitting the factor 1/2, (13.110) represents the total interaction energy. It would be incorrect to add any interaction of charges in B with potentials produced by A. We now proceed to write the interaction UAB in terms of a multipole expansion. This can be done in two ways: (i) The potential φB (r) in (13.110) is expanded in a Taylor series around the center r A , involving derivatives of the potential at r A . (ii) The source terms qj (rj ) in (13.111) are expanded in a Taylor series around the center r B . Both methods lead to nearly equivalent multipole deﬁnitions, with subtle diﬀerences that we subsequently discuss. Combined they result in a description of the interaction between two charge clouds as a sum of interactions between multipoles. The expansions are only convergent when riA < rAB , resp. rjB < rAB , which is fulﬁlled when the charge distributions do not overlap.

13.8.1 Expansion of the potential In the following we shall use a notation with greek subscripts α, β, . . . for cartesian components of vectors and tensors. The components of a radiusvector r are indicated by xα , etc. We use the Einstein summation convention: summation over repeated indices is assumed. Thus μα Eα means 3 α=1 μα Eα , which is equivalent to the inner vector product μ · E. We concentrate on the distribution A. The sources of the potential are external to A and we drop the superscript B for the potential to simplify the notation. Also, we take the center of the coordinate system in r A . Now we expand φ(r) in a three-dimensional Taylor series around the coordinate center, assuming that all derivatives of φ exist: ∂φ 1 ∂2φ φ(r) = φ(0) + xα (0) + xα xβ (0) ∂xα 2! ∂xα ∂xβ

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Electromagnetism

1 + xα xβ xγ 3!

∂3φ ∂xα ∂xβ ∂xγ

(0) + · · · .

(13.112)

Inserting this expansion into (13.110), we ﬁnd ∂φ 1 (2) ∂2φ (0) (1) UAB = M φ(0) + M α (0) + M αβ (0) ∂xα 2! ∂xα ∂xβ 1 (3) ∂3φ + M αβγ (0) + · · · , (13.113) 3! ∂xα ∂xβ ∂xγ (n)

is a form of the n-th multipole of the distribution, which is a where M symmetric tensor of rank n:7 (0) M = qi (monopole), (13.114) i

(1) Mα

=

qi xiα = μ (dipole),

(13.115)

qi xiα xiβ = Q (quadrupole),

(13.116)

qi xiα xiβ xiγ = O (octupole),

(13.117)

i

(2)

M αβ =

i

(3) M αβγ

=

i

etc. (hexadecapole,8 . . . ). We use an overline to denote this form of the multipole moments, as they are not the deﬁnitions we shall ﬁnally adopt. The quadrupole moments and higher multipoles, as deﬁned above, contain parts that do not transform as a tensor of rank n. They are therefore reducible. From the quadrupole we can separate the trace tr Q = Qαα = Qxx + Qyy + Qzz ; this part is a scalar as it transforms as a tensor of rank 0. Deﬁning Q as the traceless tensor Q = 3Q − ( tr Q) 1, qj (3xjα xjβ − rj2 δαβ ), Qαβ =

(13.118) (13.119)

j

the quadrupolar term in the energy expression (13.113) becomes 1 ∂2φ ∂2φ 1 Qαβ = Qαβ + ( tr Q)∇2 φ. 2 ∂xα ∂xβ 6 ∂xα ∂xβ 7

8

(13.120)

A real tensor in 3D space is deﬁned by its transformation property under a rotation R of the (cartesian) coordinate system, such that tensorial relations are invariant for rotation. For a rank-0 tensor t (a scalar) the transformation is t = t; for a rank-1 tensor v (a vector) the = R transformation is v = Rv or vα αβ vβ . For a rank-2 tensor T the transformation is T = RTRT or Tαβ = Rαγ Rβ δ Tγ δ , etc. The names di-, quadru-, octu- and hexadecapole stem from the minimum number of charges needed to represent the pure general n-th multipole.

13.8 Multipole expansion

357

Since the potential has no sources in domain A, the Laplacian of φ is zero, and the second term in this equation vanishes. Thus the energy can just as well be expressed in terms of the traceless quadrupole moment Q. The latter is a pure rank-2 tensor and is deﬁned by only ﬁve elements since it is symmetric and traceless. It can always be transformed to a diagonal tensor with two elements by rotation (which itself is deﬁned by three independent elements as Eulerian angles). The octupole case is similar, but more complex. The tensor has 27 elements, but symmetry requires that any permutation of indices yields the same tensor, leaving ten diﬀerent elements. Partial sums that can be eliminated because they multiply with the vanishing Laplacian of the potential are of the form Oxxx + Oyyx + Ozzx = 0.

(13.121)

There are thee such equations that are not related by symmetry, thus leaving only seven independent elements. These relations are fulﬁlled if the octupole as deﬁned in (13.135) is corrected as follows: Oαβγ = qj [5xjα xjβ xjγ − rj2 (xjα δβγ + xjβ δγα + xjγ δαβ )]. (13.122) j

This is the rank-3 equivalent of the traceless rank-2 tensor. The energy expression (13.113) can also be written in tensor notation as UAB = qφ(0) − μ · E − 16 Q:∇E −

1 30 O:∇∇E

+ ···,

(13.123)

where the semicolon denotes the scalar product deﬁned by summation over all corresponding indices. E is the electric ﬁeld vector, ∇E its gradient and ∇∇E the gradient of its gradient.9

13.8.2 Expansion of the source terms Now consider the source charge distribution qj (r j ). The potential at an arbitrary point r (such as rA) is given by φ(r) =

qj 1 . 4πε0 |r − r j |

(13.124)

j

When the point r is outside the charge distribution, i.e., |r| > |r j | for any j, the right-hand side can be expanded with the Taylor expansion of |r − r j |−1 9

Note that we do not write a dot, as ∇ · E means its divergence or scalar product; we write ∇∇ and not ∇2 , as the latter would indicate the Laplacian, which is again a scalar product.

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Electromagnetism

in terms of powers of rj /r. The general Taylor expansion in three dimensions is ∂f 1 ∂2 (a) + xα xβ (a) + · · · . (13.125) f (a + x) = f (a) + xα ∂xα 2! ∂xα ∂xβ So we shall need derivatives of 1/r. These are not only needed for expansions of this type, but also come in handy for ﬁelds, ﬁeld gradients, etc. It is therefore useful to list a few of these derivatives: ∂ 1 xα = − 3, (13.126) ∂xα r r xα xβ 1 ∂2 1 = 3 5 − 3 δαβ , (13.127) ∂xα ∂xβ r r r xα xβ xγ 3 1 ∂3 = −15 + 5 (xα δβγ + xβ δγα + xγ δαβ ),(13.128) 7 ∂xα ∂xβ ∂xγ r r r 4 xα xβ xγ xδ 15 1 ∂ = 105 − 7 (xα xβ δγδ + xβ xγ δδα 9 ∂xα · · · ∂xδ r r r +xγ xδ δαβ + xδ xα δβγ + xα xγ δβδ + xβ xδ δγα ) 3 (13.129) + 5 (δαβ δγδ + δβγ δδα + δαγ δβδ ). r These derivatives are in vector notation: ∇ 1r , ∇∇ 1r , ∇∇∇ 1r , ∇∇∇∇ 1r . Expanding φ(r) in inverse powers of r is now a matter of substituting these derivatives into the Taylor expansion of |r−r j |−1 . Using the same deﬁnitions for the multipoles as in (13.114) to (13.117), we obtain: δαβ 1 (2) 3xα xβ (0) 1 (1) xα 4πε0 φ(r) = M + M α 3 + M αβ − 3 r r 2! r5 r xα xβ xγ 1 (3) 3 + M αβγ 15 − 5 (xα δβγ + xβ δγα + xγ δαβ ) 3! r7 r +···. (13.130) The terms in this sum are of increasing order in r−n , and represent the potentials of monopole, dipole, quadrupole and octupole, respectively. (l) Instead of M we can also use the traceless deﬁnitions, because the trace do not contribute to the potential. For example, the trace part of the quadrupole (which is a constant times the unit matrix) leads to a con tribution 3α=1 r−3 (3x2α − r2 ) = 0. Therefore instead of (13.130) we can write: δαβ 1 (0) 1 (1) 1 (2) 3xα xβ 4πε0 φ(r) = M + 3 Mα xα + Mαβ − 3 r r 6 r5 r

13.8 Multipole expansion

1 (3) M 10 αβγ +..., +

with M (0) =

5xα xβ xγ 1 − 5 (xα δβγ 7 r r

+ xβ δγα + xγ δαβ )

359

(13.131)

qj = q (monopole),

(13.132)

qj xjα = μ (dipole),

(13.133)

qj (3xjα xjβ − rj2 δαβ) = Q (quadrupole),

(13.134)

j

Mα(1) =

j

(2)

Mαβ =

j

(3)

Mαβγ =

qj [5xjα xjβ xjγ − rj2 (xjα δβγ + xjβ δγα + xjγ δαβ )]

j

= O (octupole).

(13.135)

Expressed in terms of the derivative tensors, the potential reads: 1 1 1 1 1 q − μ · ∇ + Q:∇∇ − O:∇∇∇ + · · · . (13.136) r r 6 r 30 r The multipole deﬁnitions obviously also apply to continuous charge distributions, when the summations are replaced by integration over space and the charges qj are replaced by a charge density. These are the deﬁnitions (in cartesian coordinates) that we shall adopt for the multipole moments. The reader should be aware that there is no consensus on the proper deﬁnition of multipole moments and diﬀerent deﬁnitions are used in the literature.10 Not only the deﬁnitions may diﬀer, but also the choice of center is important for all multipoles beyond the lowest non-zero multipole. If the total charge (monopole) is non-zero, the dipole moment depends on the choice of origin; the dipole moment will vanish if the center of charge i qi r i / i qi is chosen as the center of the expansion. Likewise the quadrupole moment depends on the choice of origin for dipolar molecules, etc. Another elegant and popular expansion of the source term is in terms of spherical harmonics Ylm (θ, φ). These are functions expressed in polar and azimuthal angles; for use in simulations they are often less suitable than their cartesian equivalents. For higher multipoles they have the advantage of being restricted to the minimum number of elements while the cartesian 4πε0 φ(r) =

10

Our deﬁnition corresponds to the one used by Hirschfelder et al. (1954) and to the one in general use for the deﬁnition of nuclear electric quadrupole moments in NMR spectroscopy (see, e.g., Schlichter, 1963). In molecular physics the quadrupole is often deﬁned with an extra factor 1/2, corresponding to the Legendre polynomials with l = 2, as in the reported quadrupole moment of the water molecule by Verhoeven and Dymanus (1970). The deﬁnition is not always properly reported and the reader should carefully check the context.

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Electromagnetism

z θ r

θj α rj

x

y

ϕj

ϕ

Figure 13.4 The source is at r j = (rj , θj , φj ) and the potential is determined at r = (r, θ, φ). The angle between these two vectors is α.

tensors contain superﬂuous elements (as 27 cartesian tensor components against the minimum of seven irreducible components for the octupole). On the other hand, for numerical computations it is generally advisable not to use higher multipoles on a small number of centers at all, but rather use lower multipoles (even only monopoles) on a larger number of centers, in order to avoid complex expressions. For example, the computation of a force resulting from dipole–dipole interaction requires the gradient of a dipole ﬁeld, which involves a rank-3 tensor already; this is generally as far as one is prepared to go. Instead of including quadrupoles, one may choose a larger number of centers instead, without loss of accuracy. The expansion in spherical harmonics is based on the fact that the inverse distance 1/|r − r j | is a generating function for Legendre polynomials Pl (cos α), where α is the angle between r and r j (see Fig. 13.4): r 2 rj l rj j cos α + )−1/2 = Pl (cos α), r r r ∞

(1 − 2

(13.137)

l=0

where the ﬁrst four Legendre polynomials are given by Pl0 in (13.140) to (13.149) below. These Legendre polynomials of the cosine of an angle α between two directions characterized by polar and azimuthal angles (θ, φ) and (θj , φj ) can

13.8 Multipole expansion

361

subsequently be expanded by the spherical harmonics addition theorem:11 Pl (cos α) =

l (l − |m|)! m Y (θ, φ)Yl−m (θj φj ), (l + |m|)! l

(13.138)

m=−l

where |m|

Ylm (θ, φ) = Pl

(cos θ)eimφ |m|

(13.139)

are the associated Legendre

are the spherical harmonic functions and Pl functions.12 For l ≤ 3 these functions are: l = 0 : P00 (cos θ) = 1, l=1 : : l=2 : : : l=3 : : : :

P10 (cos θ) P11 (cos θ) P20 (cos θ) P21 (cos θ) P22 (cos θ) P30 (cos θ) P31 (cos θ) P32 (cos θ) P33 (cos θ)

(13.140)

= cos θ,

(13.141)

= sin θ,

(13.142)

=

1 2 2 (3 cos θ

− 1),

(13.143)

= 3 sin θ cos θ,

(13.144)

2

(13.145)

= 3 sin θ, = =

5 2 3 2

cos3 θ

− cos θ, 3 2

sin θ(5 cos2 θ

− 1),

(13.146) (13.147)

2

= 15 sin θ cos θ,

(13.148)

3

(13.149)

1 (l − |m|)! m m M Y (θ, φ), (l + |m|)! l l

(13.150)

= 15 sin θ.

The result is 4πε0 φ(r) =

∞ l l=0 m=−l

rl+1

where Mm l are the 2l + 1 components of the l-th spherical multipole: qj rjl Yl−m (θj , φj ). (13.151) Mm l = j

These spherical harmonic deﬁnitions are related to the cartesian tensor definitions of (13.132) to (13.135). 11

12

We use simple non-normalized spherical harmonics. Our deﬁnition of the spherical multipole moments corresponds to Hirschfelder et al. (1965). Deﬁnitions in the literature may diﬀer as to the normalization factors and sign of the functions for odd m. See, e.g., Weisstein (2005), Abramowitz and Stegun (1965) and Press et al. (1992). See, e.g., Jahnke and Emde (1945), who list Legendre functions up to l = 6 and associated functions up to l = 4.

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Electromagnetism

13.9 Potentials and ﬁelds in non-periodic systems Given a set of charges, the calculation of potentials, ﬁelds, energies and forces by summation of all pairwise interactions is a problem of N 2 complexity that easily runs out of hand for large systems. The use of a cut-oﬀ radius reduces the problem to order-N , but produces large and often intolerable artefacts for the fairly long-ranged Coulomb forces. For gravitational forces, lacking the compensation of sources with opposite sign, cut-oﬀs are not allowed at all. Eﬃcient methods that include long-range interactions are of two types: (i) hierarchical and multipole methods, employing a clustering of sources for interactions at longer distances; and (ii) grid methods, essentially splitting the interaction into short- and long-range parts, solving the latter by Poisson’s equation, generally on a grid. The second class of methods are the methods of choice for periodic system, which are treated in detail in the next section. They can in principle also be used for non-periodic systems – and still with reasonable eﬃciency – by extending the system with periodic images. But also without periodicity the same method of solution can be used when the Poisson equation is solved on a grid with given boundary conditions, possibly employing multigrid methods with spatial resolution adjusted to local densities. As we emphasize molecular simulations where the long-range problem concerns Coulomb rather than gravitational forces, we shall not further consider the hierarchical and “fast multipole” methods, which are essential for astrophysical simulations and are also used in molecular simulation,13 but have not really survived the competition with methods described in the next section. Whether the fast multipole methods may play a further role in molecular simulation, is a matter of debate (Board and Schulten, 2000).

13.10 Potentials and ﬁelds in periodic systems of charges In periodic systems (see Section 6.2.1) the Coulomb energy of the charges is given by (see (13.42)): UC = 12 i qi (φ(ri ) − φself ), i ∈ unit cell, (13.152) 13

The basic articles on hierarchical and fast multipole methods are Appel (1985), Barnes and Hut (1986) and Greengard and Rokhlin (1987). Niedermeier and Tavan (1994) and Figueirido et al. (1997) describe the use of fast multipole methods in molecular simulations. It is indicated that these methods, scaling proportional to N , are computationally more eﬃcient than lattice summation techniques for systems with more than about 20 000 particles.

13.10 Potentials and ﬁelds in periodic systems of charges

with φ(r) =

1 4πε0

j

n1 ,n2 ,n3 ∈Z

j

n∈Z

363

qj |r − r j − n1 a − n2 b − n3 c|

qj 1 , = 4πε0 |r − rj − Tn| 3

(13.153)

where T is the transformation matrix from relative coordinates in the unit cell to cartesian coordinates (see (6.3) on page 143), i.e., a matrix of which the columns are the cartesian base vectors of the unit cell a, b, c. The last line of (13.153) is in matrix notation; the meaning of |x| is (xT x)1/2 . Note that the displacements can be either subtracted (as shown) or added in (13.153). The self-energy contains the diverging interaction of qi with itself, but not with the images of itself; the images are to be considered as diﬀerent particles as in a crystal. The interaction of a charge with its images produces zero force, as for every image there is another image at equal distance in the opposite direction; thus the interaction energy of each charge with its own images is a constant, which diverges with the number of periodic images considered. In order to avoid the divergence we may assume that every charge has a homogeneous charge distribution of equal magnitude but opposite sign associated with it. If the total charge in a unit cell vanishes, i.e., for electroneutral systems, the homogeneous background cancels and need not be invoked. The direct sum of Coulomb terms is only conditionally convergent (i.e., the convergence depends on the sequence of terms in the summation) and converges very slowly. For an eﬃcient evaluation of the lattice energies and forces it is necessary to split the Coulomb potential into a short-range part that can be directly evaluated by summation in real space, and a long-range part that can be eﬃciently computed by solving Poisson’s equation. The easiest way to accomplish this is to consider each (point) charge as a sum of two charge distributions (see Fig. 13.5): qi δ(r − r i ) = qi [δ(r − r i ) − w(r − r i )] + qi w(r − r i ),

(13.154)

where w(r) = w(r) is an isotropic spread function which decreases smoothly and rapidly with distance and integrates to 1 over space: ∞ w(r)4πr2 dr = 1. (13.155) 0

For the time being we do not specify the spread function and derive the equations in a general form. Subsequently two speciﬁc spread functions will be considered.

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Electromagnetism

δ(x)

δ(x)-w(x) x

w(x) rc

=

+

Figure 13.5 A point charge with δ-function distribution (left) is split up into a distribution with short-range potential (middle) and a distribution with long-range potential (right) by a smooth charge-spread function w(r).

The total Coulomb energy is split into two contributions: 1 1 qi φsi + qi φli , i ∈ unit cell. UC = UCs + UCl = 2 2 i

(13.156)

i

Note that we did not split the energy into the sum of energies of the two charge distributions and their interactions, which would require four terms. Each of the contributions should include the self-energy correction. In addition there is a contribution to the energy as a result of the net dipole moment of the unit cell, treated in Section 13.10.5. 13.10.1 Short-range contribution For the short-range contributions UCs to the energy we can write: (13.157) UCs = 12 i qi φsi , i ∈ unit cell. φsi =

1 qj ϕs (rijn ), 4πε0 n

(13.158)

j

def

where the prime in the sum means exclusion of j = i for n = 0, r ijn = ri − rj − Tn and ϕs is a potential function related to the spread function: ∞ ∞ def s 1 dr 2 dr 4πr2 w(r ). (13.159) ϕ (r) = r r r The force F si on particle i due to the short-range potential equals the charge qi times the electric ﬁeld E s (r i ) = −(∇φs (r))r i : F si = −qi (∇φs (r))r i =

r ijn qi qj f s (rijn ) , 4πε0 rijn 3 j

where

fs

(13.160)

n∈Z

is a force function related to the spread function: dϕs (r) 1 ∞ def s f (r) = − w(r ) 4πr2 dr . = 2 dr r r

(13.161)

13.10 Potentials and ﬁelds in periodic systems of charges

365

One may also evaluate the force on particle i from taking minus the gradient of the total short-range energy (13.158). Although the expression for the energy contains a factor 12 , particle number i occurs twice in the summation, and one obtains the same equation (13.160) as above. Note that, by omitting j = i for n = 0 from the sum, the short-range terms are corrected for the short-range part of the self-energy. Similarly, Coulomb interactions between speciﬁed pairs can be omitted from the short-range evaluation, if so prescribed by the force ﬁeld. Usually, Coulomb interactions are omitted between atoms that are ﬁrst or second (and often modiﬁed for third) neighbors in a covalent structure because other bond and bond-angle terms take care of the interaction. When the spread function is such that the potentials and forces are negligible beyond a cut-oﬀ distance rc , which does not exceed half the smallest box size, the sums contain only one nearest image of each particle pair, which can best be evaluated using a pair list that also contains a code for the proper displacement to obtain the nearest image for each pair. 13.10.2 Long-range contribution The long-range contribution expressed as an explicit particle sum

1 1 l s φ (r i ) = qj − ϕ (rijn ) (13.162) 4πε0 rijn n j

converges very slowly because of the 1/r nature of the function. The longrange potential can be eﬃciently evaluated by solving Poisson’s equation (see (13.47)): −ε0 ∇2 φl (r) = ρl (r) = qi w(r − ri − Tn). (13.163) i

n∈Z3

The solution is equivalent to (13.162), except that no restrictions, such as j = i for n = 0 or any other speciﬁed pairs, can be included. The Poisson solution therefore contains a self-energy part (which is a constant, given the unit cell base vectors and spread function), that must be subtracted separately. If Coulomb interactions between speciﬁed pairs must be omitted, their contribution included in the long-range interaction must be subtracted. The charge distribution is periodic, and so must be the solution of this equation. The solution is determined up to any additive periodic function satisfying the Laplace equation ∇2 φ = 0, which can only be a constant if continuity at the cell boundaries is required. The constant is irrelevant. There are several ways to solve the Poisson equation for periodic systems,

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Electromagnetism

including iterative relaxation on a lattice (see Press et al., 1992), but the obvious solution can be obtained in reciprocal space, because the Laplace operator then transforms to a simple multiplication. We now proceed to formulate this Fourier solution. First deﬁne the discrete set of wave vectors k = 2π(m1 a∗ + m2 b∗ + m3 c∗ ),

(13.164)

with m1 , m2 , m3 ∈ Z, which enumerate the Fourier terms, and a∗ , b∗ , c∗ the reciprocal lattice vectors. See Section 12.9 on page 331 for a description of the reciprocal lattice and corresponding Fourier transforms. For the evaluation of the scalar product k · r it is generally easier to use the relative coordinates (ξ, η, ζ) of r: k · r = 2π(m1 ξ + m2 η + m3 ζ),

(13.165)

with ξ = r · a∗ , etc. Now construct the Fourier transform (often called structure factors) of the ensemble of point charges: def qj e−ik·r j , j ∈ unit cell. (13.166) Qk = j

Since we have broadened the point charges with the spread function, the charge density ρl (r) (13.163) is the convolution of the point charge distribution and the spread function. The Fourier transform Pkl of ρl (r) therefore is the product of the charge structure factors and the Fourier transform Wk of the spread function: 1 l def ρl (r)e−ik·r dr (13.167) Pk = V V (13.168) = Qk Wk , where Wk

1 w(r + Tn)e−ik·(r +Tn) dr V n V 1 w(r)e−ik·r dr = V all space π 1 ∞ dr dθ 2πr2 w(r) sin θ e−ikr cos θ = V 0 0 1 ∞ sin kr dr = 4πrw(r) V 0 k

def

=

(13.169)

(13.170)

Here V means integration over one unit cell, and we have used the fact that the spread function is isotropic. We see that Wk depends only on the

13.10 Potentials and ﬁelds in periodic systems of charges

367

absolute value k of k. The validity of (13.168) can easily be checked by evaluating (13.167) using (13.163). The Poisson equation (13.163) in reciprocal space reads −k 2 ε0 Φlk = −Pkl ,

(13.171)

and thus Φlk =

Qk Wk ; k = 0. ε0 k 2

(13.172)

Note that k = 0 must not be allowed in this equation and Φ0 is therefore not deﬁned; indeed for electroneutral systems Q0 = 0. Electroneutrality therefore is required; if the system is charged the potential does not converge and electroneutrality must be enforced by adding a homogeneous background charge of opposite sign. This enforces Q0 = 0. The real-space potential φl (r) follows up to a constant by Fourier transformation: Qk Wk (13.173) Φlk eik·r = eik·r . φl (r) = ε0 k 2 k =0

k =0

The total energy can be expressed in terms of a sum over wave vectors 1 1 −2 l UCl = qi φ(r i ) = k Qk Q−k Wk − Uself . (13.174) 2 2ε0 i

k =0

The self-energy contained in the long-range energy is a constant given by the j = i, n = 0 part of (13.162): 1 2 l = qi lim [r−1 − ϕs (r)] (13.175) Uself r→0 4πε0 i

ϕs

see (13.159)). Similarly, the interaction energy between excluded (for pairs ij for which the long-range energy must be corrected is qi qj −1 l Uexcl = [r − ϕs (rij )]. (13.176) 4πε0 ij The long-range force on particle i can be evaluated from the gradient of the potential: Qk Wk ikeik·r i . (13.177) F li = qi E l (r i ) = −qi (∇φl (r))r i = −qi ε0 k 2 k =0

The sum is real since for every k-term there is a complex conjugate −kterm. The self-energy term does not produce a force; the ij exclusion term produces a long-range force on particle i (and the opposite force on j): qi qj −2 r ij [rij − f s (rij )] (13.178) F li,excl = 4πε0 rij

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Electromagnetism

(for f s see (13.161)) which should be subtracted from the long-range force evaluation.

13.10.3 Gaussian spread function In the special case a Gaussian distribution is chosen for the spread function, the solution is expressed in sums of analytical functions and the classical Ewald summation is obtained (Ewald, 1921). The advantage of a Gaussian function is that its Fourier transform is also a Gaussian function, and both the real-space and reciprocal-space functions taper oﬀ quickly and can be restricted to a limited range. The Gaussian function contains an inverse width parameter β (the variance of the Gaussian distribution is 1/2β 2 ); if β is small, the spread function is wide and the real-space summation has many terms. The reciprocal functions, on the other hand, then decrease rapidly with increasing k. For large β the inverse is true. Therefore β can be tuned for the best compromise, minimizing the computational eﬀort for a given error margin. Noting that x 2 def 2 e−u du, (13.179) erf (x) = √ π 0 ∞ 2 2 def e−u du, (13.180) erfc (x) = 1 − erf (x) = √ π x we summarize the relevant functions: w(r) = w(r) = (13.159) (13.161) (13.170)

β 3 −(βr)2 e , π 3/2

1 erfc (βr), r 1 2 β −β 2 r2 e , f s (r) = 2 erfc (βr) + √ r πr k2 1 exp − 2 . Wk = V 4β

ϕs (r) =

(13.181) (13.182) (13.183) (13.184)

The explicit expressions for the energies and forces are given below. A prime above a sum means that the self-term i = j for n = 0 and the excluded pairs (i, j, n) ∈ exclusion list are excluded from the sum.

(13.158)

UCs

erfc (βrijn ) 1 1 = qi qj , 4πε0 2 rijn n i,j

(13.185)

13.10 Potentials and ﬁelds in periodic systems of charges

(13.160)

UCl =

1 1 1 k2 ik·(r i −r j ) qi qj e exp − 2ε0 V 2 k2 4β 2

(13.175) (13.176)

(13.160)

l Uself = l = Uexcl

F si

1 4πε0 1 4πε0

−

l Uexcl ,

i

369

k =0

i,j

l −Uself

qi2

(13.186) 2β √ , π

i,j,n∈exclusionlist

(13.187) qi qj

erf (βrijn ) , rijn

(13.188)

erfc (βr) qi 2β −β 2 r2 , = qj + √ e 4πε0 r2 r π n j

(13.189) r = rijn = |r i − r j − Tn|, 2 qi ik k exp − 2 eik·(r i −r j ) , (13.190) ((13.177) F li = − 2 ε0 V k 4β j k qi qj 2 β −β 2 r2 r erf (βr) l l e , −√ (13.178) F i,excl = −F j,excl = 4πε0 r2 r πr r = r i − r j − Tn, (i, j, n) ∈ exclusion list. (13.191) The exclusion forces F li,excl must be subtracted from the long-range force F li calculated from (13.190). There is no force due the self-energy contained in the long-range energy. Figure 13.6 shows the Gaussian spread function and the corresponding short- and long-range potential functions, the latter adding up to the total potential 1/r.

13.10.4 Cubic spread function The Gaussian spread function is by no means the only possible choice.14 In fact, a spread function that leads to forces which go smoothly to zero at a given cut-oﬀ radius rc and stay exactly zero beyond that radius, have the advantage above Gaussian functions that no cut-oﬀ artifacts are introduced in the integration of the equations of motion. Any charge spread function that is exactly zero beyond rc will produce a short-range force with zero value and zero derivative at rc . In addition we require that the Fourier transform rapidly decays for large k in order to allow eﬃcient determination of the long-range forces; this implies a smooth spread function. A discontinuous function, and even a discontinuous derivative, will produce wiggles in the 14

Berendsen (1993) lists a number of choices, but does not include the cubic spread function.

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Electromagnetism

3 2.5

ϕs(r)

2

r −1

1.5

r −1 − ϕs(r)

1 w(r)

0.5 0.5

1

1.5

2

2.5

3 βr

Figure 13.6 Functions for the Ewald sum: w(r) is proportional to the Gaussian spread function; ϕs (r) and r−1 − ϕs (r) are the short-range and long-range potential functions, adding up to the total Coulomb interaction r−1 .

Fourier transform. The following cubic polynomial fulﬁlls all requirements; the force function even has a vanishing second derivative, allowing the use of higher-order integration algorithms. The functions are all analytical, although tabulation is recommended for eﬃcient implementation. Figure 13.7 shows spread and potential functions for the cubic spread function comparable to Fig. 13.6. Figure 13.8 shows the Fourier transform Wk of both the Gaussian and the cubic spread functions. The cubic charge spread function is 3r2 2r3 15 (1 − + 3 ) for r < rc , 4πrc3 rc2 rc = 0 for r ≥ rc ,

w(r) =

and its Fourier transform (13.170) is given by

8 8 90 5 (1 − 2 ) cos κ − sin κ + 2 , Wκ = 4 κ V κ κ κ

(13.192)

(13.193)

where κ = krc . The short-range force function (13.161) is f s (r) =

1 5r 9r3 5r4 − + 5 − 6 for r < rc , r2 rc3 rc rc = 0 for r ≥ rc ,(13.194)

13.10 Potentials and ﬁelds in periodic systems of charges

371

3 2.5

ϕs(r)

2

r −1

1.5

r −1 − ϕs(r)

1 0.5

w(r) 0.25

0.5

0.75

1

1.25

1.5 r/rc

Figure 13.7 Functions for the cubic spread function: w(r) is proportional to the spread function; the potential functions are as in Fig. 13.6, but scaled by 2 to make them comparable.

and the short-range potential function (13.159) is 9 5r2 9r4 r5 1 − + 3 − 5 + 6 for r < rc , r 4rc 2rc 4rc rc = 0 for r ≥ rc . (13.195)

13.10.5 Net dipolar energy Special attention needs to be given to the energetic eﬀects of a net non-zero dipole moment, as has been carefully done by de Leeuw et al. (1980).15 The problem is that Coulomb lattice sums over unit cells with non-vanishing total dipole moment converge only conditionally, i.e., the sum depends on the sequence of terms in the summation. Consider summation over a chunk of matter containing a (very large, but not inﬁnite) number of unit cells. The total dipole moment of the chunk of matter is proportional to the volume of the chunk. The Coulomb energy, given by the summed dipolar interactions, now depends on the shape of the chunk and on its dielectric environment. For example, in a ﬂat disc perpendicular to the dipole moment, the interaction is unfavorable (positive), but in a long cylinder parallel to the dipole moment the interaction is favorable (negative). In a sphere of radius R with cubic unit cells the interactions sum to zero, but there will be a reaction ﬁeld ERF 15

See also Caillol (1994), Essmann et al. (1995) and Deserno and Holm (1998a).

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Electromagnetism

1 0.8

0.04 0.02 0 -0.02

0.6

7.5

10

12.5

15

17.5

7.5

10

12.5

15

17.5

0.4 0.2 0 2.5

5

20 krc

Figure 13.8 Fourier transforms of the cubic (solid line) and Gaussian (dashed line) spread functions. For the Gaussian transform β was set to 2/rc . The inset magniﬁes the tails.

due to the polarizability of the medium in which the sphere is embedded (see (13.83) on page 347): ERF =

μtot 2(εr − 1) , 4πε0 R3 2εr + 1

(13.196)

where εr is the relative dielectric constant of the medium. The energy per unit cell −μtot ERF /(2N ) (where N is the number of unit cells in the sphere) in the reaction ﬁeld can now be written as URF = −

μ2 2(εr − 1) , 6ε0 V 2εr + 1

(13.197)

where μ is the unit cell dipole moment and V the unit cell volume. This term does not depend on the size of the system since the R3 proportionality in the volume just cancels the R−3 proportionality of the reaction ﬁeld. For lower multipoles (i.e., for the total charge) the energy diverges, and the system is therefore required to be electroneutral; for higher multipoles the lattice sum converges unconditionally so that the problem does not arise. It is clear that the boundary conditions must be speciﬁed for periodic systems with non-vanishing total dipole moment. The system behavior, especially the ﬂuctuation of the dipole moment, will depend on the chosen boundary conditions. A special case is the tin-foil or metallic boundary condition, given by εr = ∞, which is equivalent to a conducting boundary.

13.10 Potentials and ﬁelds in periodic systems of charges

373

Applied to a sphere, the RF energy per unit cell then becomes URF = −

μ2 6ε0 V

(spherical tin-foil b.c.).

(13.198)

Since the question of the boundary condition did not come up when solving for the long-range Coulomb interaction, leading to (13.174), one wonders whether this equation silently implies a speciﬁc boundary condition, and if so, which one. By expanding exp(±ik · r) in powers of k, we see that Qk Q−k = (k · μ)2 + O(k 4 ), while Wk = (1/V ) + O(k 2 ). The term (k · μ)2 equals 13 μ2 k 2 when averaged over all orientations of the dipole moment. Thus the energy term k −2 Qk Q−k Wk /(2ε0 ) equals −μ2 /(6ε0 V ) + O(k 2 ), which is exactly the dipolar energy for the tin-foil boundary conditions. The conclusion is that application of the equations for the Coulomb energy, as derived here based on a splitting between short- and long-range components, and consequently also for the Ewald summation, automatically imply tin-foil boundary conditions. If one wishes to exert spherical boundary conditions corresponding to a dielectric environment with relative dielectric constant εr rather than conducting boundary conditions, an extra term making up the diﬀerence between (13.197) and (13.198) must be added to the computed energy. This extra term is 1 μ2 Udipole = . (13.199) 2ε0 V (2εr + 1) This term is always positive, as the tin-foil condition (for which the correction is zero) provides the most favorable interaction. In a vacuum environment (εr = 1) it is more unfavorable to develop a net dipole moment, and in a dipolar ﬂuid with ﬂuctuating net dipole moment, the net dipole moment is suppressed compared to tin-foil boundary conditions. The most natural boundary condition for a dipolar ﬂuid would be a dielectric environment with a dielectric constant equal to the actual dielectric constant of the medium. 13.10.6 Particle–mesh methods The computational eﬀort of the Ewald summation scales as N 2 with the number of charges N and becomes prohibitive for large systems.16 Fast Fourier transforms (FFT)17 are computationally attractive although they 16

17

With optimized truncation of the real and reciprocal sums (Perram et al., 1988) a N 3/2 -scaling can be accomplished. The computation can also be made considerably faster by using tabulated functions (see Chapter 19). See Section 12.7 of Chapter 12 on page 324 for details on fast Fourier transforms.

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Electromagnetism

restrict the spatial solutions to lattice points. Interpolation is then needed to obtain the energies and forces acting on charges. They scale as N log N and form the long-range methods of choice, e.g., as implemented in the particle– mesh–Ewald (PME) method of Darden et al. (1993) who use a Gaussian spread function and a Lagrange interpolation, or – preferably – the more accurate smooth particle–mesh–Ewald (SPME) method of Essmann et al. (1995), who use a B-spline interpolation.18 The advantage of using B-spline interpolation is that the resulting potential function is twice continuously diﬀerentiable if the order of the spline is at least four; smooth forces can therefore be immediately obtained from the diﬀerentiated potential. With Lagrange interpolation the interpolated potential is only piecewise diﬀerentiable and cannot be used to derive the forces. The SPME works as follows, given a system of N charges qi at positions ri within a unit cell of base vectors a.b, c: • Choose three integers K1 , K2 , K3 that subdivide the unit cell into small, reasonably isotropic, grid cells. Choose an appropriate Ewald parameter β (see (13.181)), a cutoﬀ-radius rc for the short-range interaction, which should not exceed half the length of the smallest base vector, and a cut-oﬀ radius in reciprocal space. Choose the order p of the B-spline interpolation, which should be at least 4 (cubic spline). A cutoﬀ of 4 times the √standard deviation of the Gaussian spread function implies that βrc = 2 2. A grid size a/K1 , b/K2 , c/K3 of about 0.3/β and a reciprocal cut-oﬀ of 1.5β would be reasonable for a start. The optimal values depend on system size and density; they should be adjusted for optimal computational eﬃciency, given a desired overall accuracy. • Compute the structure factors using the exponential spline interpolation as explained in Chapter 19, using (19.87) and (19.88) on page 554. Note that for odd order p the value m = ±K/2 must be excluded For actual programs the reader is referred to the original authors. A full description of the SPME algorithm can be found in Griebel et al. (2003). These methods use a splitting between short-range and long-range potentials and solve the long-range potential on a grid; they are really variants of the PPPM particle–particle particle–mesh) method developed earlier by Hockney and Eastwood (1988). In the PPPM method the charges are distributed over grid points; the Poisson equation is solved and the potentials and ﬁelds are interpolated, using optimized local functions that minimize the total error. Deserno and Holm (1998a) have compared the accuracies and 18

For details on B-splines see Chapter 19, Section 19.7, on page 548.

13.10 Potentials and ﬁelds in periodic systems of charges

375

eﬃciencies of various methods and evaluated the error in a PPPM scheme in a second paper (1998b). It seems that SPME methods have not yet been applied to other charge spread functions than Gaussian ones, although short-range force functions that go exactly and smoothly to zero at the cut-oﬀ radius would have the advantage of avoiding cut-oﬀ noise in the short-range forces. A suitable candidate would be the cubic function, discussed on page 369.

13.10.7 Potentials and ﬁelds in periodic systems of charges and dipoles Certain force ﬁelds describe charge distributions not only with a set of charges, but also with a set of dipoles, or even higher multipoles. The dipoles may be permanent ones, designed to describe the charge distribution with a smaller number of sites. The may also be induced dipoles proportional to the local ﬁeld, as in certain types of polarizable force ﬁelds. In the latter case the induced dipoles are determined in an iterative procedure until consistency, or they may be considered as variables that minimize a free energy functional. In all cases it is necessary to determine potentials and ﬁelds, and from those energies and forces, from a given distribution of charges and dipoles. The methods described above, splitting interactions into short- and longrange parts, can be extended to include dipolar sources as well. One must be aware that such extensions considerably complicate the computation of energies and forces, as the dipolar terms involve higher derivatives than are required for charges. It could well be advantageous to avoid dipoles – and certainly higher multipoles – if the problem at hand allows formulation in terms of charges alone. Here we shall review the methods in order to give the reader a ﬂavor of the additional complexity, referring the reader to the literature for details. Ewald summations including multipolar sources have been worked out by Smith (1982); based on this work, Toukmaji et al. (2000) extended PME methods to dipolar sources. The eﬀect of adding dipole moments μi to the sources qi is that qi is replaced by qi +μi ·∇i , which has consequences for the structure factors as well as for the short- and long-range force and energy terms. Consider the energy U12 between two charges q1 at r 1 and q2 at r 2 : U12 = q1 Φ(r 1 ) =

1 1 q1 q2 , 4πε0 r12

(13.200)

376

Electromagnetism

where r12 = |r 1 − r 2 |.19 When dipoles are present this modiﬁes to (cf (13.123)) U12 = q1 Φ(r 1 ) − μ1 · E(r 1 ) = (q1 + μ1 · ∇1 )Φ(r 1 ),

(13.201)

with the potential given by (see (13.136)) 4πε0 Φ(r 1 ) =

q2 1 1 − μ2 · ∇1 = (q2 + μ2 · ∇2 ) . r12 r12 r12

(13.202)

The interaction thus changes from q1 q2 /r12 to U12 = (q1 + μ1 · ∇1 )(q2 + μ2 · ∇2 )

1 . r12

(13.203)

We note that this represents the total electrostatic interaction; for the energy of polarizable systems one should add the energy it costs to create the induced dipole moments, which is a quadratic form in μ (like i μ2i /2αi ) for the case of linear polarizability. This replacement works through all equations; for example, the structure factor Qk (13.166) now becomes (qj + μj · ∇j )e−ik·r j Qk = j

=

(qj − 2πiμj · k)e−ik·r j , j ∈ unit cell,

(13.204)

j

with consequences for the spline interpolation procedure. The reader is referred to Toukmaji et al. (2000) for details. Exercises 13.1

13.2

13.3

19

Consider an electron as a (classical) sphere with homogeneous charge distribution. What would its radius be when the total ﬁeld energy equals the relativistic rest energy mc2 ? What is the amplitude in vacuum of E and B in a circular laser beam of 50 mW monochromatic radiation (λ = 632.8 nm), when the beam has a diameter of 2 mm? Consider an inﬁnite planar surface with vacuum on one side and a dielectric medium with relative dielectric constant εr on the other side, with a charge q situated on the vacuum side at a distance d from the plane. Show that the following potential satisﬁes the boundary conditions (13.56): on the vacuum side the direct Coulomb

See the beginning of Section 13.8 on page 353 for a discussion of charge interactions and where a factor 2 should or should not appear.

Exercises

13.4

13.5

377

potential of the charge plus the vacuum potential of an image charge qim = −q(εr − 1)/(εr + 1) at the mirror position; on the medium side the direct Coulomb potential of the charge, divided by a factor εeﬀ . Express εeﬀ in εr . Compare the exact solvation free energy of two charges q1 and q2 , both at a distance d from the planar surface that separates vacuum and medium (εr ) as in Exercise 13.3, and separated laterally by a distance r12 , with the generalized Born expression (13.103) using Still’s expression (13.105). Verhoeven and Dymanus (1970) have measured the quadrupole moment of D2 O. They report the values: θa = 2.72(2), θb = −0.32(3), θc = −2.40, in 10−26 esu.cm2 , for the diagonal components in a coordinate system with its origin in the center-of-mass, where a and b are in the plane of the molecule and b is in the direction of the molecular symmetry axis. They use the following traceless deﬁnition of the quadrupole moment: 1 θαβ = ρ(r)[3xα xβ − r2 δαβ ] dr. 2 From these data, derive the quadrupole moment Qαβ as deﬁned in (13.119) on page 356, expressed in “molecular units” e nm2 (see Table 8 on page xxv), and in a coordinate system with its origin in the position of the oxygen atom. Use the following data for the transformation: OD-distance: 0.09584 nm, DOD-angle: 104◦ 27 , dipole moment: 1.85 Debye, oxygen mass: 15.999 u, deuterium mass: 2.014 u. An esu (electrostatic unit) of charge equals 3.335 64 × 10−10 C; the elementary charge e equals 4.8032 × 10−10 esu. Give the accuracies as well.

14 Vectors, operators and vector spaces

14.1 Introduction A vector we know as an arrow in 3D-space with a direction and a length, and we can add any two vectors to produce a new vector in the same space. If we deﬁne three coordinate axes, not all in one plane, and deﬁne three basis vectors e1 , e2 , e3 along these axes, then any vector v in 3D-space can be written as a linear combination of the basis vectors: v = v1 e1 + v2 e2 + v3 e3 .

(14.1)

v1 , v2 , v3 are the components of v on the given basis set. These components form a speciﬁc representation of the vector, depending on the choice of basis vectors. The components are usually represented as a matrix of one column: ⎞ ⎛ v1 v = ⎝ v2 ⎠ . (14.2) v3 Note that the matrix v and the vector v are diﬀerent things: v is an entity in space independent of any coordinate system we choose; v represents v on a speciﬁc set of basis vectors. To stress this diﬀerence we use a diﬀerent notation: italic bold for vectors and roman bold for their matrix representations. Vectors and basis vectors need not be arrows in 3D-space. They can also represent other constructs for which it is meaningful to form linear combinations. For example, they could represent functions of one or more variables. Consider all possible real polynomials f (x) of the second degree, which can be written as f (x) = a + bx + cx2 ,

(14.3)

where a, b, c can be any real number. We could now deﬁne the functions 1, x, 379

380

Vectors, operators and vector spaces

and x2 as basis vectors (or basis functions) and consider f (x) as a vector with components (a, b, c) on this basis set. These vectors also live in a real 3D-space R3 . 14.2 Deﬁnitions Now we wish to give more general and a bit more precise deﬁnitions, without claiming to be mathematically exact. • A set of elements, called vectors, form a vector space V over a scalar ﬁeld F when: (i) V is an Abelian group under the sum operation +; (ii) for every v ∈ V and every a ∈ F : av ∈ V; (iii) for every v, w ∈ V and a, b ∈ F: a(v + w) = av + bw (a + b)v = av + bv (ab)v = a(bv) 1v = v 0v = 0 A scalar ﬁeld is precisely deﬁned in set theory, but for our purpose it suﬃces to identify F with the set of real numbers R or the set of complex numbers C. An Abelian group is a set of elements for which a binary operation + (in this case a summation) is deﬁned such that v +w = w +v is also an element of the set, in which an element 0 exists for which v + 0 = 0 + v, and in which for every v an element −v exists with v + (−v) = 0. • A vector space is n-dimensional if n vectors e1 , . . . , en exist, such that every element v ∈ V can be written as v = ni=1 vi ei . The n vectors must be linearly independent, i.e., no non-zero set of numbers c1 , c2 , . . . , cn exists for which ni=1 ci ei = 0. The vectors e1 , . . . , en form a basis of V. • A vector space is real if F = R and complex if F = C. • A vector space is normed if to every v a non-negative real number ||v|| is associated (called the norm), such that for every v, w ∈ V and every complex number c: (v) ||cv|| = |c|||v||; (vi) ||v + w|| ≤ ||v|| + ||w||; (vii) ||v|| > 0 for v = 0. • A vector space is complete if:

14.3 Hilbert spaces of wave functions

381

(viii) for every series v n with limm,n→∞ ||v m − v n || = 0 there exists a v such that limm,n→∞ ||v − v n || = 0. Don’t worry: all vector spaces we encounter are complete. • A Banach space is a complete, normed vector space. • A Hilbert space H is a Banach space in which a scalar product or inner product is deﬁned as follows: to every pair v, w a complex number is associated (often denoted by (v, w) or v|w), such that for every u, v, w ∈ H and every complex number c: (ix) (x) (xi) (xii) (xiii)

cv|w = c∗ v|w; u + v|w = u|w + v|w; v|w = w|v∗ ; v|v > 0 if v = 0; ||v|| = v|v1/2 .

• Two vectors are orthogonal if v|w = 0. A vector is normalized if ||v|| = 1. A set of vectors is orthogonal if all pairs are orthogonal and the set is orthonormal if all vectors are in addition normalized.

14.3 Hilbert spaces of wave functions We consider functions ψ (it is irrelevant what variables these are functions of) that can be expanded in a set of basis functions φn : cn φn , (14.4) ψ= n

where cn are complex numbers. The functions may also be complex-valued. We deﬁne the scalar product of two functions as (φ, ψ) = φ|ψ = φ∗ ψ dτ, (14.5) where the integral is over a deﬁned volume of the variables τ . The norm is now deﬁned as ||ψ|| = ψ ∗ ψ dτ. (14.6) These deﬁnitions comply with requirements (viii) - (xii) of the previous section, as the reader can easily check. Thus the functions are vectors in a Hilbert space; the components c1 , . . . , cn form a representation of the vector ψ which we shall denote in matrix notation by the one-column matrix c.

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Vectors, operators and vector spaces

The basis set {φn } is orthonormal if φn |φm = δnm . It is not mandatory, but very convenient, to work with orthonormal basis sets. For nonorthonormal basis sets it is useful to deﬁne the overlap matrix S: Snm = φn |φm .

(14.7)

The representation c of a normalized function satisﬁes1 c† c = c∗n cn = 1

(14.8)

n

on an orthogonal basis set; on an arbitrary basis set c† Sc = 1. 14.4 Operators in Hilbert space An operator acts on a function (or vector) to transform it into another function (or vector) in the same space. We restrict ourselves to linear operators which transform a function into a linear combination of other functions and ˆ denote operators by a hat, as A: ˆ ψ = Aψ.

(14.9)

An operator can be represented by a matrix on a given orthonormal basis set {φn }, transforming the representation c of ψ into c of ψ by an ordinary matrix multiplication c = Ac, where

Anm

ˆ m = = φn |A|φ

(14.10) ˆ m dτ. φ∗n Aφ

(14.11)

Proof Expanding ψ on an orthonormal basis set φm and applying (14.9) we have: ˆ = ˆ m. ψ = cm φm = Aψ cm Aφ m

m

φ∗n

Now left-multiply by and integrate over coordinates to form the scalar products ˆ m = cm φn |φm = cm φn |A|φ Anm cm , m

m

m

or cn = (Ac)n . 1

With the superscript † we denote the hermitian conjugate, which is the transpose of the complex conjugate: (A† )nm = A∗mn . This is the usual notation in physics and chemistry, but in mathematical texts the hermitian conjugate is often denoted by ∗.

14.4 Operators in Hilbert space

383

ˆ The eigenvalue equation for an operator A: ˆ = λψ, Aψ

(14.12)

now becomes on an orthonormal basis set an eigenvalue equation for the matrix A: Ac = λc.

(14.13)

Solutions are eigenvectors c and eigenvalues λ. If the basis set is not orthonormal, the equation becomes Ac = λSc.

(14.14)

Hermitian operators form an important subclass of operators. An operator Aˆ is hermitian if ˆ = g|Af ˆ ∗ , f |Ag or

∗

ˆ dτ = f Ag

(Aˆ∗ f ∗ )g dτ.

(14.15)

(14.16)

Hermitian operators have real expectation values (f = g = ψ) and real ˆ = λf → λ = λ∗ ). The operators of physically meaneigenvalues (f = g; Af ingful observables are hermitian. The matrix representation of a hermitian operator is a hermitian matrix A = A† (f = φn , g = φm ). Not only do hermitian operators have real eigenvalues, they also have orthogonal eigenfunctions for non-degenerate (diﬀerent) eigenvalues. The eigenfunctions within the subspace corresponding to a set of degenerate eigenvalues can be chosen to be orthogonal as well,2 and all eigenfunctions may be normalized: The eigenfunctions of a hermitian operator (can be chosen to) form an orthonormal set. ˆ Proof Let λn , ψn be eigenvalues and eigenfunctions of A: ˆ n = λn ψ n . Aψ Then

∗

ˆ m dτ = λm ψ Aψ 2

ψn∗ ψm dτ,

If φ1 and φ2 are two eigenfunctions of the same (degenerate) eigenvalue λ, then any linear combination of φ1 and φ2 is also an eigenfunction.

384

Vectors, operators and vector spaces

and

∗ ˆ ∗ ∗ ∗ ψn dτ. ( ψm Aψn dτ ) = λn ψm

† ∗ When A = A∗ then for n = m : λn = λn → λ is real; for m = n and λm = λn : ψn ψm dτ = 0.

ˆ B] ˆ of two operators is deﬁned as The commutator [A, ˆ B] ˆ = AˆB ˆ − B, ˆ A, ˆ [A,

(14.17)

ˆ commute if their commutator is zero. If two and we say that Aˆ and B operators commute, they have the same set of eigenvectors.

14.5 Transformations of the basis set It is important to clearly distinguish operators that act on functions (vectors) in Hilbert space, changing the vector itself, from coordinate transformations which are operators acting on the basis functions, thus changing the representation of a vector, without touching the vector itself. Consider a linear coordinate transformation Q changing a basis set {φn } into a new basis set {φn }: φn = Qin φi . (14.18) i

Let A be the representation of an operator Aˆ on {φn } and A its representation on {φn }. Then A and A relate as A = Q† AQ.

(14.19)

Proof Consider one element of A and insert (14.18): ˆ = (A )nm = φn |A|φ Q∗in Qjm Aij = (Q† AQ)nm . m ij

If both basis sets are orthonormal, then the transformation Q is unitary.3 Proof Orthonormality implies that φn |φm = δnm . 3

A transformation (matrix) U is unitary if U† = U−1 .

14.6 Exponential operators and matrices

Since φn |φm =

Q∗in Qjm φi |φj =

ij

it follows that

385

(Q† )ni Qim ,

i

(Q† )ni Qim = δnm ,

i

or Q† Q = 1, which implies that Q is unitary. The representations c and c of a vector are related as c = Qc ,

−1

†

c = Q

c,

c = Q c,

(14.20) (14.21) (14.22)

where (14.22) is only valid for unitary transformations. Let Aˆ be a hermitian operator with eigenvalues λ1 , λ2 , . . . and with orthonormal eigenvectors c1 , c2 , . . .. Then, if we construct a matrix U with columns formed by the eigenvectors, it follows that AU = ΛU,

(14.23)

where Λ is the diagonal matrix of eigenvalues. Since all columns of U are orthonormal, U is a unitary matrix, and thus U† AU = Λ.

(14.24)

In other words: U is exactly the coordinate transformation that diagonalizes A.

14.6 Exponential operators and matrices ˆ = eAˆ , e.g., as formal We shall often encounter operators of the form exp(A) solutions of ﬁrst-order diﬀerential equations. The deﬁnition is ˆ

eA =

∞ 1 ˆk A . k!

(14.25)

k=0

Exponential matrices are similarly deﬁned. From the deﬁnition it follows that ˆ Aˆ = eAˆ A, ˆ Ae

(14.26)

386

Vectors, operators and vector spaces

and ˆ

ˆ

ˆ

eA (f + g) = eA f + eA g.

(14.27)

ˆ is exp(A): The matrix representation of the operator exp(A)

∞ ∞ 1 1 k ˆ n|A |m = (Ak )nm = (eA )nm . n|e |m = k! k! ˆ A

k=0

(14.28)

k=0

The matrix element (exp A)nm is in general not equal to exp(Anm ), unless A is a diagonal matrix Λ = diag(λ1 , . . .): (eA )nm = eλn δnm .

(14.29)

ˆ or exp(A) transforms just like any From the deﬁnition follows that exp(A) other operator under a unitary transformation: U† eA U = ˆ

∞ ∞ 1 † ˆk 1 ˆ ˆ k = eU† AU U A U= (U† AU) . k! k! k=0

(14.30)

k=0

This transformation property is true not only for unitary transformations, but for any similarity transformation Q−1 AQ. Noting that the trace of a matrix is invariant for a similarity transformation, it follows that A λn (14.31) λn = exp( tr A). det(e ) = Πn e = exp n

Some other useful properties of exponential matrices or operators are A −1 e = e−A , (14.32) and d At e = AeAt = eAt A (t is a scalar variable). dt

(14.33)

Generally, exp(A + B) = exp(A) exp(B), unless A and B commute. If A and B are small (proportional to a smallness parameter ε), the ﬁrst error term is of order ε2 and proportional to the commutator [A, B]: eε(A+B) = eεA eεB − 12 ε2 [A, B] + O(ε3 ), εB εA

= e

e

+

1 2 2 ε [A, B]

+

O(ε3 ).

(14.34) (14.35)

We can approximate exp(A + B) in a series of approximations, called the Lie–Trotter–Suzuki expansion. These approximations are quite useful for the design of stable algorithms to solve the evolution in time of quantum or classical systems (see de Raedt, 1987, 1996). The basic equation, named the

14.6 Exponential operators and matrices

387

Trotter formula after Trotter (1959), but based on earlier ideas of Lie (see Lie and Engel, 1888), is m . (14.36) e(A+B) = lim eA/m eB/m m→∞

Let us try to solve the time propagator U (τ ) = e−i(A+B)τ ,

(14.37)

where A and B are real matrices or operators. The ﬁrst-order solution is obviously U1 (τ ) = e−iAτ e−iBτ + O(τ 2 ).

(14.38)

U1† (τ ) = eiBτ eiAτ = U1−1 (τ ),

(14.39)

Since

the propagator U1 (τ ) is unitary. Suzuki (1991) gives a recursive recipe to derive higher-order products for the exponential operator. Symmetric products are special cases, leading to algorithms with even-order precision. For second order precision Suzuki obtains U2 (τ ) = e−iBτ /2 e−iAτ e−iBτ /2 + O(τ 3 ).

(14.40)

Higher-order precision is obtained by the recursion equation (for symmetric products; m ≥ 2) U2m (τ ) = [U2m−2 (pm τ )]2 U2m−2 ((1 − 4pm )τ )[U2m−2 (pm τ )]2 + O(τ 2m+1 ), (14.41) with 1 pm = . (14.42) 1/(2m−1) 4−4 For fourth-order precision this works out to U4 (τ ) = U2 (pτ )U2 (pτ )U2 ((1 − 4p)τ )U2 (pτ )U2 (pτ ) + O(τ 5 ),

(14.43)

with p=

1 = 0.4145. 4 − 41/3

(14.44)

All of the product operators are unitary, which means that algorithms based on these product operators – provided they realize the unitary character of each term – are unconditionally stable. The following relation is very useful as starting point to derive the behavior of reduced systems, which can be viewed as projections of the complete

388

Vectors, operators and vector spaces

system onto a reduced space (Chapter 8). For any pair of time-independent, ˆ we can write non-commuting operators Aˆ and B t ˆ ˆ ˆ ˆ ˆ B)τ ˆ ˆ (A+ eA(t−τ ) Be dτ. (14.45) e(A+B)t = eAt + 0

Proof First write ˆ

ˆ

ˆ

ˆ e(A+B)t = eAt Q(t),

(14.46)

so that ˆ (A+ ˆ ˆ ˆ = e−At e B)t . Q(t)

By diﬀerentiating (14.46), using the diﬀerentiation rule (14.33), we ﬁnd ˆ ˆ ˆ ˆ dQ ˆ B)t ˆ ˆ At ˆ (A+ Q(t) + eAt , = Ae (Aˆ + B)e dt and using the equality ˆ ˆ ˆ B)t ˆ ˆ B)t ˆ ˆ At ˆ (A+ ˆ (A+ = Ae , (Aˆ + B)e Q(t) + Be

we see that ˆ dQ ˆ ˆ (A+ ˆ B)t ˆ = e−At Be . dt ˆ Hence, by integration, and noting that Q(0) = 1: t ˆ ˆ (A+ ˆ B)τ ˆ ˆ =1+ e−Aτ Be dτ. Q(t) 0

ˆ in (14.46) yields the desired expression. Inserting this Q There are two routes to practical computation of the matrix exp(A). The ﬁrst is to diagonalize A : Q−1 AQ = Λ and construct eA = Q diag (eλ1 , eλ2 , . . .) Q−1 .

(14.47)

For large matrices diagonalization may not be feasible. Then in favorable cases the matrix may be split up into a sum of block-diagonal matrices, each of which is easy to diagonalize, and the Trotter expansion applied to the exponential of the sum of matrices. It may also prove possible to split the operator into a diagonal part and a part that is diagonal in reciprocal space, and therefore solvable by Fourier transformation, again applying the Trotter expansion.

14.6 Exponential operators and matrices

389

The second method4 is an application of the Caley–Hamilton relation, which states that every n × n matrix satisﬁes its characteristic equation An + a1 An−1 + a2 An−2 + . . . + an−1 A + an 1 = 0.

(14.48)

Here a1 , . . . an are the coeﬃcients of the characteristic or eigenvalue equation det(A − λ1) = 0, which is a n-th degree polynomial in λ: λn + a1 λn−1 + a2 λn−2 + . . . + an−1 λ + an = 0.

(14.49)

Equation (14.49) is valid for each eigenvalue, and therefore for the diagonal matrix Λ; (14.48) then follows by applying the similarity transformation QΛQ−1 . According to the Caley–Hamilton relation, An can be expressed as a linear combination of Ak , k = 0, . . . , n − 1, and so can any Am , m ≥ n. Therefore, the inﬁnite sum in (14.25) can be replaced by a sum over powers of A up to n − 1: eA = μ0 1 + μ1 A + · · · + μn−1 An−1 .

(14.50)

The coeﬃcients μi can be found by solving the system of equations μ0 + μ1 λk + μ2 λ2k + · · · + μn−1 λn−1 = exp(λk ), k = 1, . . . , n k

(14.51)

(which follows immediately from (14.50) by transforming exp(A) to diagonal form). In the case of degenerate eigenvalues, (14.51) are dependent and the superﬂuous equations must be replaced by derivatives: = exp(λk ) μ1 + 2μ2 λk + · · · + (n − 1)μn−1 λn−2 k

(14.52)

for a doubly degenerate eigenvalue, and higher derivatives for more than doubly degenerate eigenvalues.

14.6.1 Example of a degenerate case Find the exponential matrix for ⎛

⎞ 0 1 0 A = ⎝ 1 0 0 ⎠. 0 0 1 According to the Caley–Hamilton relation, the exponential matrix can be expressed as exp(A) = μ0 1 + μ1 A + μ2 A2 . 4

See, e.g., Hiller (1983) for the application of this method in system theory.

390

Vectors, operators and vector spaces

Note that the eigenvalues are +1, +1, −1 and that A2 = I. The equations for μ are (because of the twofold degeneracy of λ1 the second line is the derivative of the ﬁrst) μ0 + μ1 λ1 + μ2 λ21 = exp(λ1 ), μ1 + 2μ2 λ1 = exp(λ1 ), μ0 + μ1 λ3 + μ2 λ23 = exp(λ3 ). Solving for μ we ﬁnd 1 1 μ0 = μ2 = (e + ), 4 e 1 1 μ1 = (e − ), 2 e which yields the exponential matrix ⎞ ⎛ e + 1/e e − 1/e 0 1 eA = ⎝ e − 1/e e + 1/e 0 ⎠ . 2 0 0 2e The reader is invited to check this solution with the ﬁrst method.

14.7 Equations of motion In this section we consider solutions of the time-dependent Schr¨ odinger equation, both in terms of the wave function and its vector representations, and in terms of the expectation values of observables.

14.7.1 Equations of motion for the wave function and its representation The time-dependent Schr¨ odinger equation ∂ i ˆ Ψ(r, t) = − HΨ(r, t) ∂t

(14.53)

reads as vector equation in Hilbert space on a stationary orthonormal basis set: i (14.54) c˙ = − Hc. In these equations the Hamiltonian operator or matrix may itself be a function of time, e.g., it could contain time-dependent external potentials.

14.7 Equations of motion

These equations can be formally solved as i t ˆ Ψ(r, t) = exp − H(t ) dt Ψ(r, 0), 0 i t c(t) = exp − H(t ) dt c(0), 0

391

(14.55) (14.56)

which reduce in the case that the Hamiltonian does not depend explicitly on time to i ˆ (14.57) Ψ(r, t) = exp − Ht Ψ(r, 0), i (14.58) c(t) = exp − Ht c(0). These exponential operators are propagators of the wave function in time, to be written as ˆ (t)Ψ(r, 0), Ψ(r, t) = U

(14.59)

c(t) = U(t)c(0).

(14.60)

The propagators are unitary because they must keep the wave function normalized at all times: c† c(t) = c(0)† U† Uc(0) = 1 for all times only if U† U = 1. We must agree on the interpretation of the role of the time in the exponent: the exponential operator is time-ordered in the sense that changes at later times act subsequent to changes at earlier times. This means that, for t = t1 + t2 , where t1 is ﬁrst, followed by t2 , the operator factorizes as i ˆ i ˆ i ˆ (14.61) exp − Ht = exp − Ht2 exp − Ht1 . Time derivatives must be interpreted as ˆ (t + dt) = − i dt H(t) ˆ U ˆ (t), U

(14.62)

ˆ and H ˆ do not commute. even when U

14.7.2 Equation of motion for observables The equation of motion for the expectation A of an observable with operˆ ator A, A = Ψ|A|Ψ,

(14.63)

392

Vectors, operators and vector spaces

is given by i ˆ ˆ d A = [H, A] + dt Proof d dt

ˆ dt = i Ψ AΨ ∗

∂A ∂t

.

ˆ dτ + ∂A − i ˆ ∗ Ψ∗ )AΨ (H ∂t

(14.64)

ˆ dτ. Ψ∗ AˆHΨ

ˆ is hermitian: Because H ˆ AΨ ˆ dτ, ˆ ∗ Ψ∗ )AΨ ˆ dτ = Ψ∗ H (H and (14.64) follows. Instead of solving the time-dependence for several observables separately by (14.64), it is more convenient to solve for c(t) and derive the observables from c. When ensemble averages are required, the method of choice is to use the density matrix, which we shall now introduce.

14.8 The density matrix Let c be the coeﬃcient vector of the wave function Ψ(r, t). on a given orthonormal basis set. We deﬁne the density matrix ρ by ρnm = cn c∗m ,

(14.65)

ρ = cc† .

(14.66)

or, equivalently,

The expectation value of an observable A is given by c∗m φ∗m Aˆ cn φn dτ = ρnm Amn = (ρA)nn A = m

n

n,m

(14.67)

n

so that we obtain the simple equation5 A = tr ρA.

(14.68)

So, if we have solved c(t) then we know ρ(t) and hence A(t). The evolution of the density matrix in time can also be solved directly from its equation of motion, called the Liouville–von Neumann equation: ρ˙ = 5

i [ρ, H].

The trace of a matrix is the sum of its diagonal elements.

(14.69)

14.8 The density matrix

393

Proof By taking the time derivative of (14.66) and applying (14.54), we see that i i ˙ † + cc˙ † = − Hcc† + c(Hc)† . ρ˙ = cc Now (Hc)† = c† H† = c† H because H is hermitian, so that i i i ρ˙ = − Hρ + ρH = [ρ, H].

This equation also has a formal solution: i i ρ(t) = exp − Ht ρ(0) exp + Ht ,

(14.70)

where, if H is time-dependent, H(t) in the exponent is to be replaced by t ) dt . H(t 0 Proof We prove that the time derivative of (14.70) is the equation of motion (14.69): i i i ρ˙ = − H exp − Ht ρ(0) exp + Ht (14.71) i i i (14.72) = + exp − Ht ρ(0)H exp + Ht i (14.73) = [ρ, H]. Here we have used the fact that H and exp − i Ht commute. The density matrix transforms as any other matrix under a unitary coordinate transformation U: ρ = U† ρU.

(14.74)

On a basis on which H is diagonal (i.e., on a basis of eigenfunctions of ˆ : Hnm = En δnm ) the solution of ρ(t) is H i ρnm (t) = ρnm (0) exp (Em − En )t , (14.75) implying that ρnn is constant.

394

Vectors, operators and vector spaces

14.8.1 The ensemble-averaged density matrix The density matrix can be averaged over a statistical ensemble of systems without loss of information about ensemble-averaged observables. This is in contrast to the use of c(t) which contains a phase factor and generally averages out to zero over an ensemble. In thermodynamic equilibrium (in the canonical ensemble) the probability of a system to be in the n-th eigenstate with energy En is proportional to its Boltzmann factor: 1 Pn = e−βEn , (14.76) Q where β = 1/kB T and Q=

e−βEn

(14.77)

n

is the partition function (summed over all quantum states). On a basis set ˆ in which H is diagonal, of eigenfunctions of H, 1 −βH e , Q Q = tr e−βH ,

ρeq =

(14.78) (14.79)

implying that oﬀ-diagonal elements vanish, which is equivalent to the assumption that the phases of ρnm are randomly distributed over the ensemble (random phase approximation). But (14.78) and (14.79) are also valid after any unitary coordinate transformation, and thus these equations are generally valid on any orthonormal basis set. 14.8.2 The density matrix in coordinate representation The ensemble-averaged density matrix gives information on the probability of quantum states φn , but it does not give direct information on the probability of a conﬁguration of the particles in space. In the coordinate representation we deﬁne the equilibrium density matrix as a function of (multiparticle) spatial coordinates r: ρ(r, r ; β) = φ∗n (r)e−βEn φn (r ). (14.80) n

This is a square continuous “matrix” of ∞ × ∞ dimensions. The trace of ρ is tr ρ = ρ(r, r; β) dr, (14.81)

14.8 The density matrix

395

which is equal to the partition function Q. A product of such matrices is in fact an integral, which is itself equal to a density matrix: (14.82) ρ(r, r 1 ; β1 )ρ(r 1 , r ; β2 ) dr 1 = ρ(r, r ; β1 + β2 ), as we can check by working out the l.h.s.: ∗ −β1 En ∗ −β2 Em φn (r)e φn (r 1 ) φm (r 1 )e φm (r ) dr 1 n

=

m

φ∗n (r)e−β1 En −β2 Em φm (r )

φn (r 1 )φ∗m (r 1 ) dr 1

n,m

=

φ∗n (r)e−(β1 +β2 )En φn (r ) = ρ(r, r ; β1 + β2 ).

n

A special form of this equality is ρ(r, r ; β) = ρ(r, r 1 ; β/2)ρ(r 1 , r ; β/2) dr 1 ,

(14.83)

which can be written more generally as (14.84) ρ(r, r ; β) = ρ(r, r 1 ; β/n)ρ(r 1 , r 2 ; β/n) . . . ρ(r n−1 , r ; β/n) dr 1 , . . . , dr n−1 . Applying this to the case r = r , we see that Q = tr ρ (14.85) = ρ(r, r 1 ; β/n)ρ(r 1 , r 2 ; β/n) . . . ρ(r n−1 , r; β/n) dr dr 1 , . . . , dr n−1 . Thus the partition function can be obtained by an integral over density matrices with the “high temperature” β/n; such density matrices can be approximated because of the small value in the exponent. This equality is used in path integral Monte Carlo methods to incorporate quantum distributions of “heavy” particles into simulations.

15 Lagrangian and Hamiltonian mechanics

15.1 Introduction Classical mechanics is not only an approximation of quantum mechanics, valid for heavy particles, but historically it also forms the basis on which quantum-mechanical notions are formed. We also need to be able to describe mechanics in generalized coordinates if we wish to treat constraints or introduce other ways to reduce the number of degrees of freedom. The basis for this is Lagrangian mechanics, from which the Hamiltonian description is derived. The latter is not only used in the Schr¨ odinger equation, but forms also the framework in which (classical) statistical mechanics is developed. A background in Lagrangian and Hamiltonian mechanics is therefore required for many subjects treated in this book. After the derivation of Lagrangian and Hamiltonian dynamics, we shall consider how constraints can be built in. The common type of constraint is a holonomic constraint that depends only on coordinates, such as a bond length constraint, or constraints between particles that make them behave as one rigid body. An example of a non-holonomic constraint is the total kinetic energy (to be kept at a constant value or at a prescribed time-dependent value). We shall only give a concise review; for details the reader is referred to text books on classical mechanics, in particular to Landau and Lifschitz (1982) and to Goldstein et al. (2002). There are several ways to introduce the principles of mechanics, leading to Newton’s laws that express the equations of motion of a mechanical system. A powerful and elegant way is to start with Hamilton’s principle of least action as a postulate. This is the way chosen by Landau and Lifshitz (1982). 397

398

Lagrangian and Hamiltonian mechanics

15.2 Lagrangian mechanics Consider a system described by n degrees of freedom or coordinates q = q1 , . . . , qn (not necessarily the 3N cartesian coordinates of N particles) that evolve in time t. A function L(q, q, ˙ t) exists with the property that the action t2 def S = L(q, q, ˙ t) dt (15.1) t1

is minimal for the actual path followed by q(t), given it is at coordinates q(t1 ) and q(t2 ) at the times t1 and t2 . L is called the Lagrangian of the system. This principle (see the proof below) leads to the Lagrange equations d ∂L ∂L − =0 (i = 1, · · · , n). (15.2) dt ∂ q˙i ∂qi The Lagrangian is not uniquely determined by this requirement because any (total) time derivative of some function of q and t will have an action independent of the path and can therefore be added to L. Proof We prove (15.2). The variation of the action (15.1), when the path between q(t1 ) and q(t2 ) is varied (but the end points are kept constant), must vanish if S is a minimum: t2 ∂L ∂L δS = δq + δ q˙ dt = 0. ∂q ∂ q˙ t1 Partial integration of the second term, with δ q˙ = dδq/dt and realizing that δq = 0 at both integration limits t1 and t2 because there q is kept constant, converts this second term to t2 d ∂L − dt. δq dt ∂ q˙ t1 Now

t2

δS = t1

∂L d − ∂q dt

∂L ∂ q˙

δq dt = 0.

Since the variation must be zero for any choice of δq, (15.2) follows. For a free particle with position r and velocity v the Lagrangian L can only be a function of v 2 if we assume isotropy of space-time, i.e., that mechanical laws do not depend on the position of the space and time origins and on the orientation in space. In fact, from the requirement that the particle

15.3 Hamiltonian mechanics

399

behavior is the same in a coordinate system moving with constant velocity, it follows1 that L must be proportional to v 2 . For a system of particles interacting through a position-dependent potential V (r 1 , . . . , r N ), the following Lagrangian: L(r, v) =

N

1 2 2 mi v i

− V,

(15.3)

i=1

yields Newtons equations of motion mv˙i = −

∂V , ∂r i

(15.4)

as the reader can easily verify by applying the Lagrange equations of motion (15.2). 15.3 Hamiltonian mechanics In many cases a more appropriate description of the equations of motions in generalized coordinates is obtained with the Hamilton formalism. We ﬁrst deﬁne a generalized momentum pk , conjugate to the coordinate qk from the Lagrangian as def ∂L . (15.5) pk = ∂ q˙k Then we deﬁne a Hamiltonian H in such a way that dH is a total diﬀerential in dp and dq: n def pk q˙k − L. (15.6) H = k=1

¿From this deﬁnition it follows that n ∂L ∂L pk dq˙k + q˙k dpk − dqk − dq˙k . dH = ∂qk ∂ q˙k

(15.7)

k=1

The ﬁrst and the last terms cancel, so that a total diﬀerential in dp and dq is obtained, with the following derivatives: ∂H = q˙k , ∂pk ∂H = −p˙k . ∂qk These are Hamilton’s equations of motion. 1

See Landau and Lifshitz, (1982), Chapter 1.

(15.8) (15.9)

400

Lagrangian and Hamiltonian mechanics

The reader may check that these also lead to Newton’s equations of motion for a system of particles interacting through a coordinate-dependent potential V , where 1 p2k + V (q). H(p, q) = 2 2mk n

(15.10)

k=1

In this case H is the total energy of the system of particles, composed of the kinetic energy2 K and potential energy V . If H does not depend explicitly on time, it is a constant of the motion, since ∂H ∂H dH = p˙k + q˙k dt ∂pk ∂qk k (q˙k p˙k − p˙k q˙k ) = 0. (15.11) = k

So Hamiltonian mechanics conserves the value of H, or – in the case of an interaction potential that depends on position only – the total energy. Therefore, in the latter case such a system is also called conservative.

15.4 Cyclic coordinates A coordinate qk is called cyclic if the Lagrangian does not depend on qk : ∂L(q, q, ˙ t) = 0 for cyclic qk . ∂qk

(15.12)

For the momentum pk = ∂L/∂ q˙k , conjugate to a cyclic coordinate, the time derivative is zero: p˙k = 0. Therefore: The momentum conjugate to a cyclic coordinate is conserved, i.e., it is a constant of the motion. An example, worked out in detail in Section 15.6, is the center-of-mass motion of a system of mutually interacting particles isolated in space: since the Lagrangian cannot depend on the position of the center of mass, its coordinates are cyclic, and the conjugate momentum, which is the total linear momentum of the system, is conserved. Hence the center of mass can only move with constant velocity. It is not always true that the cyclic coordinate itself is moving with constant velocity. An example is the motion of a diatomic molecule in free space, where the rotation angle (in a plane) is a cyclic coordinate. The conjugate momentum is the angular velocity multiplied by the moment of 2

We use the notation K for the kinetic energy rather than the usual T in order to avoid confusion with the temperature T .

15.5 Coordinate transformations

401

inertia of the molecule. So, if the bond distance changes, e.g., by vibration, the moment of inertia changes, and the angular velocity changes as well. A coordinate that is constrained to a constant value, by some property of the system itself or by an external action, also acts as a cyclic coordinate, because it is not really a variable any more. However, its time derivative is also zero, and such a coordinate vanishes from the Lagrangian altogether. In Section 15.8 the equations of motion for a system with constraints will be considered in detail.

15.5 Coordinate transformations Consider a transformation from cartesian coordinates r to general coordinates q: r i = r i (q1 , . . . , qn ), i = 1, . . . , N, n = 3N.

(15.13)

The kinetic energy can be written in terms of q: K=

N 1 i=1

2

mi r˙ 2 =

n N 1 ∂r i ∂r i mi · q˙k q˙l . 2 ∂qk ∂ql

(15.14)

k,l=1 i=1

This is a quadratic form that can be expressed in matrix notation:3 ˙ K(q, q) ˙ = 12 q˙ T M(q)q,

(15.15)

where Mkl =

N i=1

mi

∂r i ∂r i · . ∂qk ∂ql

(15.16)

The tensor M, deﬁned in (15.16), is called the mass tensor or sometimes the mass-metric tensor.4 The matrix M(q) is in general a function of the coordinates; it is symmetric and invertible (det M = 0). Its eigenvalues are the masses mi , each three-fold degenerate. Now we consider a conservative system L(q, q) ˙ = K(q, q) ˙ − V (q). 3

4

(15.17)

We use roman bold type for matrices, a vector being represented by a column matrix, in contrast to italic bold type for vectors. For example: v · w = vT w. The superscript T denotes the transpose of the matrix. The latter name refers to the analogy with the metric tensor gkl = i [(∂ri /∂qk ) · (∂ri /∂ql )] which deﬁnes the metric ofthe generalized coordinate system: the distance ds between q and q + dq is given by (ds)2 = kl gkl dqk dql .

402

Lagrangian and Hamiltonian mechanics

The conjugate momenta are deﬁned by pk =

∂K(q, q) ˙ = Mkl q˙l ∂qk

(15.18)

l

or ˙ p = Mq,

(15.19)

and the Lagrangian equations of motion are p˙k =

∂V ∂L 1 ∂M q˙ − = q˙ T . ∂qk 2 ∂qk ∂qk

(15.20)

¨: By inserting (15.20) into (15.19) a matrix equation is obtained for q 1 ∂Mαβ ∂V ∂Mkα q˙α q˙β , Mkl q¨l = − + − (15.21) ∂qk 2 ∂qk ∂qβ l

α,β

which has the general form ˙ M¨ q = T(q) + C(q, q),

(15.22)

where T is a generalized force or torque, and C is a velocity-dependent force that comprises the Coriolis and centrifugal forces. Apart from the fact that these forces are hard to evaluate, we are confronted with a set of equations that require a complexity of order n3 to solve. Recently more eﬃcient order-n algorithms have been devised as a result of developments in robotics. By inverting (15.19) to q˙ = M−1 p, the kinetic energy can be written in terms of p (using the symmetry of M): K = 12 (M−1 p)T M(M−1 p) = 12 pT M−1 p,

(15.23)

and the Hamiltonian becomes H = pT q˙ − L = pT M−1 p − K + V = 12 pT M−1 (q)p + V (q),

(15.24)

with the Hamiltonian equations of motion ∂H = (M−1 p)k , ∂pk ∂H 1 ∂M−1 ∂V = − pT p− . p˙k = − ∂qk 2 ∂qk ∂qk q˙k =

(15.25) (15.26)

(Parenthetically we note that the equivalence of the kinetic energy in (15.20) and (15.26) implies that ∂M −1 ∂M−1 = −M−1 M , ∂qk ∂qk

(15.27)

15.6 Translation and rotation

403

which also follows immediately from ∂MM−1 /∂qk = 0. We will use this relation in ano ther context.) The term −∂V /∂qk is a direct transformation of the cartesian forces F i = −∂V /∂r i : ∂r i ∂V = Fi · , (15.28) − ∂qk ∂qk i

and is therefore a kind of generalized force on qk . Note, however, that in general the time derivative of pk is not equal to this generalized force! Equality is only valid in the case that the mass tensor is independent of qk : if

∂V ∂r i ∂M = 0 then p˙k = − = Fi · . ∂qk ∂qk ∂qk

(15.29)

i

15.6 Translation and rotation Consider a system of N particles that interact mutually under a potential V int (r 1 , . . . , r N ), and are in addition subjected to an external potential V ext (r 1 , . . . , r N ). Homogeneity and isotropy of space dictate that neither the kinetic energy K nor the internal potential V int can depend on the overall position and orientation of the system of particles. As we shall see, these properties give a special meaning to the six generalized coordinates of the overall position and orientation. Their motion is determined exclusively by the external potential. In the absence of an external potential these coordinates are cyclic, and their conjugate moments – which are the total linear and angular momentum – will be conserved. It also follows that a general three-dimensional N -body system has no more than 3N − 6 internal coordinates.5

15.6.1 Translation Consider6 a transformation from r to a coordinate system in which q1 is a displacement of all coordinates in the direction speciﬁed by a unit vector n : dr i = n dq1 . Hence, for any i, ∂r i = n, ∂q1 5 6

∂ r˙ i = n. ∂ q˙1

(15.30)

For a system consisting of particles on a straight line, as a diatomic molecule, one of the rotational coordinates does not exist and so there will be at most 3N − 5 internal coordinates. We follow the line of reasoning by Goldstein (1980).

404

Lagrangian and Hamiltonian mechanics

Homogeneity of space implies that ∂K(q, q) ˙ = 0, ∂q1

∂V int = 0. ∂q1

(15.31)

For the momentum p1 conjugate to q1 follows p1 =

∂K 1 ∂ r˙ 2i ∂ r˙ i = mi = mi r˙ i · =n· mi v i , ∂ q˙1 2 ∂ q˙1 ∂ q˙1 i

i

(15.32)

i

which is the component of the total linear momentum in the direction n. Its equation of motion is ∂V ext ∂r i ∂V ext ext · n. (15.33) p˙1 = − =− · = Fi ∂q1 ∂r i ∂q1 i

i

So the motion is governed by the total external force. In the absence of an external force, the total linear momentum will be conserved. The direction of n is immaterial: therefore there are three independent general coordinates of this type. These are the components of the center of mass (c.o.m.) def 1 r cm = mi r i , with M = mi , (15.34) M i

i

with equation of motion ˙ = r cm

1 1 ext mi v i = F . M M i

(15.35)

i

In other words: the c.o.m. behaves like a single particle with mass M .

15.6.2 Rotation Consider a transformation from r to a coordinate system in which dq1 is a rotation of the whole body over an inﬁnitesimal angle around an axis in the direction speciﬁed by a unit vector n : dr i = n × r 1 dq1 . Hence, for any i ∂ r˙ i ∂r i = = n × ri . ∂q1 ∂ q˙1

(15.36)

Isotropy of space implies that ∂K(q, q) ˙ = 0, ∂q1

∂V int = 0. ∂q1

(15.37)

15.7 Rigid body motion

405

For the momentum p1 conjugate to q1 it follows that7 p1 =

∂K = ∂ q˙1 =

1 2

2

∂ r˙ i i mi ∂ q˙1

=

i mi v i

· (n × r i )

mi (r i × v i ) · n = L · n,

(15.38)

i

where the angular momentum of the system. is given by def mi r i × v i . L =

(15.39)

i

In general, L depends on the choice of the origin of the coordinate system, except when the total linear momentum is zero. The equation of motion of p1 is ∂V ext ext = F i · (n × r i ) = r i × F ext · n = T · n, (15.40) p˙1 = − i ∂q1 i

i

where the torque exerted on the system is given by def r i × F ext T = i .

(15.41)

i

Again, n can be chosen in any direction, so that the equation of motion for the angular momentum is ˙ = T. L (15.42) In general, T depends on the choice of the origin of the coordinate system, except when the total external force is zero. If there is no external torque, the angular momentum is a constant of the motion.

15.7 Rigid body motion We now consider the special case of a system of N particles in which the mutual interactions keeping the particles together conﬁne the relative positions of the particles so strongly that the whole system may be considered as one rigid body. We now transform r to generalized coordinates that consist of three c.o.m. coordinates rcm , three variables that deﬁne a real orthogonal 3 × 3 matrix R with determinant +1 (a proper rotation matrix) and 3N − 6 internal coordinates. The latter are all constrained to a constant value and therefore do not ﬁgure in the Lagrangian. 7

Use the vector multiplication rule a · (b × c) = (a × b) · c.

406

Lagrangian and Hamiltonian mechanics

Z

Z´

X´

k c

a

j

i

Y

b

X Y´ Figure 15.1 Body-ﬁxed coordinate system X Y Z with base vectors a, b, c, rotated with respect to a space-ﬁxed coordinate system XY Z with base vectors i, j, k.

Using a body-ﬁxed coordinate system X , Y , Z (see Fig. 15.1, the positions of the particles are speciﬁed by 3N coordinates ri , which are all constants. There are, of course, only 3N − 6 independent coordinates because six functions of the coordinates will determine the position and orientation of the body-ﬁxed coordinate system. The rotation matrix R transforms the coordinates of the i-th particle, relative to the center of mass, in the space-ﬁxed system to those in the body-ﬁxed system:8 ri = R(ri − rcm ).

(15.43)

The positions in the space-ﬁxed coordinate system X, Y, Z are given by ri = rcm + RT ri . 8

(15.44)

When consulting other literature, the reader should be aware of variations in notation that are commonly used. In particular the transformation matrix is often deﬁned as the transpose of our R, i.e., it transforms from the body-ﬁxed to space-ﬁxed coordinates, or the transformation is not deﬁned as a rotation of the system of coordinate axes but rather as the rotation of a vector itself.

15.7 Rigid body motion

407

In Section 15.6 we have seen that the c.o.m. (deﬁned in (15.34)) behaves independent of all other motions according to (15.35). We disregard the c.o.m. motion from now on, i.e., we choose the c.o.m. as the origin of both coordinate systems, and all vectors are relative to rcm . The rotation matrix transforms the components v in the space-ﬁxed system of any vector v to components v in the body-ﬁxed system as v = Rv,

v = RT v .

(15.45)

The columns of RT , or the rows of R, are the components of the three bodyﬁxed unit vectors (1,0,0), (0,1,0) and (0,0,1) in the space-ﬁxed system. They are the direction cosines between the axes of the two systems. Denoting the orthogonal unit base vectors of the space-ﬁxed coordinate system by i, j, k and those of the body-ﬁxed system by a, b, c, the rotation matrix is given by ⎞ ⎛ ⎞ ⎛ ax ay az a·i a·j a·k R = ⎝ bx by bz ⎠ = ⎝ b · i b · j b · k ⎠ . (15.46) cx cy cz c·i c·j c·k The rotational motion is described by the time-dependent behavior of the rotation matrix: ri (t) = R(t)ri .

(15.47)

Therefore we need diﬀerential equations for the matrix elements of R(t) or for any set of variables that determine R. There are at least three of those (such as the Euler angles that describe the orientation of the bodyﬁxed coordinate axes in the space-ﬁxed system), but one may also choose a redundant set of four to nine variables, which are then subject to internal constraint relations. If one would use the nine components of R itself, the orthogonality condition would impose six constraints. Expression in terms of Euler angles lead to the awkward Euler equations, which involve, for example, division by the sine of an angle leading to numerical problems for small angles in simulations. Another possibility is to use the homomorphism between 3D real orthogonal matrices and 2D complex unitary matrices,9 leading to the Caley–Klein parameters10 or to Wigner rotation matrices that are often used to express the eﬀect of rotations on spherical harmonic functions. But all these are too complex for practical simulations. There are two recommended techniques, with the ﬁrst being the most suitable one for the majority of cases: 9 10

See, e.g., Jones (1990). See Goldstein (1980).

408

Lagrangian and Hamiltonian mechanics

• The use of cartesian coordinates of (at least) two (linear system), three (planar system) or four (general 3D system) “particles” in combination with length constraints between them (see Section 15.8). The “particles” are dummy particles that have mass and fully represent the motion, while the points that are used to derive the forces (which are likely to be at the position of real atoms) now become virtual particles. • The integration of angular velocity in a principal axes system, preferably combined with the use of quaternions to characterize the rotation. This method of solution is described below.

15.7.1 Description in terms of angular velocities We know that the time derivative of the angular momentum (not the angular velocity!) is equal to the torque exerted on the body (see (15.42)), but that relation does not give us a straightforward equation for the rate of change of the rotation matrix, which is related to the angular velocity. First consider an inﬁnitesimal rotation dφ = ω dt of the body around an axis in the direction of the vector ω. For any point in the body dr = ω ×r dt and hence v = ω × r. Inserting this into the expression for the angular momentum L and using the vector relation a × (b × c) = (a · c)b − (a · b)c,

(15.48)

we obtain L =

mi r i × v i

i

=

mi r i × (ω × r)

i

=

mi [ri2 ω − r i (r i · ω)]

(15.49)

i

= Iω,

(15.50)

where I is the moment of inertia or inertia tensor, which is represented by a 3 × 3 symmetric matrix I=

i

mi (ri2 1 − ri rT i ),

(15.51)

15.7 Rigid body motion

written out as I=

i

409

⎛

⎞ −xi zi yi2 + zi2 −xi yi mi ⎝ −yi xi x2i + zi2 −yi zi ⎠ . −zi xi −zi yi x2i + yi2

(15.52)

Since I is a tensor and L and ω are vectors,11 the relation L = Iω is valid in any (rotated) coordinate system. For example, this relation is also valid for the primed quantities in the body-ﬁxed coordinates: L = Iω,

L = RL,

L = I ω ,

ω = Rω,

(15.53)

T

I = RIR .

(15.54)

However, we must be careful when we transform a vector v from a rotating coordinate system: the rotation itself produces an extra term ω × v in the time derivative of v: v˙ = RT v˙ + ω × v.

(15.55)

It is possible to relate the time derivative of the rotation matrix to the angular momentum in the body-ﬁxed system. Since, for an arbitrary vector v = RT v , the derivative is v˙ =

d T ˙ Tv, R v = RT v˙ + R dt

(15.56)

which, comparing with (15.55), means that ˙ T v = ω × v. R

(15.57)

˙ T v = RT (ω × v ) = RT Ω v , R

(15.58)

Hence

where Ω is a second-rank antisymmetric tensor: ⎞ ⎛ 0 −ωz ωy def Ω = ⎝ ωz 0 −ωx ⎠ , −ωy ωx 0

(15.59)

and thus ˙ T = RT Ω . R 11

(15.60)

Tensors and vectors are deﬁned by their transformation properties under orthogonal transformations. In fact, L and ω are both pseudovectors or axial vectors because they change sign under an orthogonal transformation with determinant −1., but this distinction with proper vectors is not relevant in our context.

410

Lagrangian and Hamiltonian mechanics

˙ are Other ways of expressing R ˙ = ΩT R, R

(15.61)

˙ T = ΩRT , R ˙ = RΩT . R

(15.62) (15.63)

Recall the equation of motion (15.42) for the angular momentum; in matrix notation in the space-ﬁxed coordinate system this equation is: d L˙ = (Iω) = T. dt

(15.64)

Only in the body-ﬁxed coordinates is I stationary and constant, so we must refer to those coordinates to avoid time dependent moments of inertia: d T R I ω = T, dt

(15.65)

or ˙ T I ω + RT I ω˙ = T, R ˙ T I ω , I ω˙ = RT − RR I ω˙ = T − Ω I ω .

(15.66)

˙ This The latter equation enables us to compute the angular accelerations ω. equation becomes particularly simple if the body-ﬁxed, primed, coordinate system is chosen such that the moment of inertia matrix I is diagonal. The equation of motion for the rotation around the principal X -axis then is ω˙ x =

− I Iyy Tx zz + ωy ωz , Ixx Ixx

(15.67)

and the equations for the y- and z-component follow from cyclic permutation of x, y, z in this equation. Thus the angular acceleration in the body-ﬁxed coordinates can be computed, provided both the angular velocity and the torque are known in the body-ﬁxed frame. For the latter we must know the rotation matrix, which means that the time-dependence of the rotation matrix must be solved simultaneously, e.g., from (15.61). The kinetic energy can best be calculated from the angular velocity in the body-ﬁxed coordinate system. It is given by K = 12 i mi r˙ 2i = 12 ω · L = 12 ω · (I · ω) = 12 ω T Iω = 12 ω T I ω = 12 3α=1 Iα ωα2 (principal axes). (15.68)

15.7 Rigid body motion

411

Proof Since r˙ i = ω × r i the kinetic energy is 1 1 mi r˙ 2i = mi (ω × r i )2 . K= 2 2 i

i

With the general vector rule (a × b) · (c × d) = (a · b)(a · b) − (a · d)(b · c), this can be written as K=

(15.69)

1 1 mi [ω 2 ri2 − (ω · r i )2 ] = ω · L, 2 2 i

using the expression (15.49) for L. Let us summarize the steps to obtain the equations for simulation of the angular motion of a rigid body: (i) Determine the center of mass r cm and the total mass M of the body. (ii) Determine the moment of inertia of the body and its principal axes. If not obvious from the symmetry of the body, then ﬁrst obtain I in an arbitrary body-ﬁxed coordinate system (see (15.52)) and diagonalize it subsequently. (iii) Assume the initial (cartesian) coordinates and velocities of the constituent particles are known in a given space-ﬁxed coordinate system. From these determine the initial rotation matrix R and angular velocity ω. (iv) Determine the forces F i on the constituent particles, and the total force F tot on the body. (v) Separate the c.o.m. motion: F tot /M is the acceleration of the c.o.m., and thus of all particles. Subtract mi F tot /M from every force F i . (vi) Compute the total torque T in the body-ﬁxed principal axes coordinate system, best by ﬁrst transforming the particle forces (after c.o.m. correction) with R to the body-ﬁxed frame, to obtain F i . Then apply F i × r i . (15.70) T = i

(vii) Determine the time derivative of ω from (15.67) and the time derivative of R from (15.61). Integrate both equations simultaneously with an appropriate algorithm. The diﬀerential equation for the rotation matrix (last step) can be cast in several forms, depending on the deﬁnition of the variables used to describe the matrix. In the next section we consider three diﬀerent possibilities.

412

Lagrangian and Hamiltonian mechanics

Z Z´ ϑ

X´ ψ

Y

ϕ X

K

Figure 15.2 Euler angles deﬁning the rotation of coordinate axes XY Z to X Y Z . For explanation see text.

15.7.2 Unit vectors A straightforward solution of the equation of motion for R is obtained by applying (15.61) directly to its nine components, i.e., to the body-ﬁxed basis vectors a, b, c. There are two possibilities: either one rotates ω to the spaceﬁxed axes and then applies (15.61): ω = RT ω ,

(15.71)

a˙ = ω × a

(15.72)

(and likewise for b and c), or one applies (15.63) term for term. Both methods preserve the orthonormality of the three basis vectors, but in a numerical simulation the matrix may drift slowly away from orthonormality by integration errors. Provisions must be made to correct such drift.

15.7.3 Euler angles The traditional description of rotational motion is in Euler angles, of which there are three and no problem with drifting constraints occur. The Euler angles do have other severe problems related to the singularity of the Euler equations when the second Euler angle ϑ has the value zero (see below). For that reason Euler angles are not popular for simulations and we shall only give the deﬁnitions and equations of motion for the sake of completeness. The Euler angles ϕ, ϑ, ψ, deﬁned as follows (see Fig. 15.2). We rotate XY Z in three consecutive rotations to X Y Z . First locate the line of

15.7 Rigid body motion

413

nodes K which is the line of intersection of the XY - and the X Y -planes (there are two directions for K; the choice is arbitrary). Then: (i) rotate XY Z around Z over an angle ϕ until X coincides with K; (ii) rotate around K over an angle ϑ until Z coincides with Z ; (iii) rotate around Z over an angle ψ until K coincides with X . Rotations are positive in the sense of a right-hand screw (make a ﬁst and point the thumb of your right hand in the direction of the rotation axis; your ﬁngers bend in the direction of positive rotation). The rotation matrix is ⎛ ⎞ cos ϕ cos ψ+ sin ϕ cos ψ+ + sin ϑ sin ψ ⎜ − sin ϕ cos ϑ sin ψ cos ϕ cos ϑ sin ψ ⎟ ⎜ ⎟ ⎜ ⎟ R = ⎜ − cos ϕ sin ψ+ (15.73) − sin ϕ sin ψ+ + sin ϑ cos ψ ⎟ . ⎜ ⎟ ⎝ − sin ϕ cos ϑ cos ψ cos ϕ cos ϑ cos ψ ⎠ sin ϕ sin ϑ − cos ϕ sin ϑ cos ϑ The equations of motion, relating the angular derivatives to the body-ﬁxed angular velocities, can be derived from (15.61). They are ⎞⎛ ⎞ ⎛ ⎞ ⎛ sin ϕ cos ϑ cos ϕ cos ϑ ϕ˙ ωx 1 sin ϑ sin ϑ ⎠⎝ ωy ⎠ . ⎝ ϑ˙ ⎠ = ⎝ cos ϕ (15.74) sin ϕ sin ϕ cos ϕ ˙ ωz ψ − sin ϑ sin ϑ 15.7.4 Quaternions Quaternions12

[q0 , q1 , q2 , q3 ] = q0 + q1 i + q2 j + q3 k are hypercomplex numbers with four real components that can be viewed as an extension of the complex numbers a + bi. They were invented by Hamilton (1844). The normalized quaternions, with i qi2 = 1, can be conveniently used to describe 3D rotations; these are known as the Euler–Rodrigues parameters, described a few years before Hamilton’s quaternions by Rodrigues (1840). Subsequently the quaternions have received almost no attention in the mechanics of molecules,13 until they were revived by Evans (1977). Because equations of motion using quaternions do not suﬀer from singularities as 12

13

See Altmann (1986) for an extensive review of quaternions and their use in expressing 3D rotations, including a survey of the historical development. Another general reference is Kyrala (1967). In the dynamics of macroscopic multi-body systems the Euler–Rodrigues parameters are wellknown, but not by the name “quaternion”. See for example Shabana (1989). They are commonly named Euler parameters; they diﬀer from the three Rodrigues parameters, which are deﬁned as the component of the unit vector N multiplied by tan( 12 φ) (see (15.79)).

414

Lagrangian and Hamiltonian mechanics

those with Euler angles do, and do not involve the computation of goniometric functions, they have become popular in simulations involving rigid bodies (Allen and Tildesley, 1987; Fincham, 1992). The unit quaternions14 1 = [1, 0, 0, 0], i = [0, 1, 0, 0], j = [0, 0, 1, 0] and k = [0, 0, 0, 1] obey the following multiplication rules ([q] is any quaternion): 1[q] = [q]1 = [q],

(15.75)

i = j = k = −1,

(15.76)

ij = −ji = k

(15.77)

2

2

2

and cyclic permutations.

A general quaternion can be considered as the combination [q0 , Q] of a scalar q0 and a vector Q with components (q1 , q2 , q3 ). The multiplication rules then imply (as the reader is invited to check) that [a, A][b, B] = [ab − A · B, aB + bA + A × B].

(15.78)

According to Euler’s famous theorem,15 any 3D rotation with ﬁxed origin can be characterized as single rotation about an axis n over an angle φ: R(φ, n). Thus four parameters φ, nx , ny , nz describe a rotation, with the constraint that |n| = 1. Note that R(−φ, −n) and R(φ, n) represent the same rotation. The Euler–Rodrigues parameters are expressions of these four parameters: [q] = [cos 12 φ, n sin 12 φ].

(15.79)

They are indeed quaternions that obey the multiplication rule (15.78). They should be viewed as rotation operators, the product [a][b] meaning the sequential operation of ﬁrst [b], then [a]. Beware that [q] rotates a vector in a given coordinate system and not the coordinate system itself, which implies that [q] is to be identiﬁed with RT and not with R. The unit quaternions now have the meaning of the identity operator ([1, 0, 0, 0]), and rotations by π about the x-, y- and z-axes, respectively. Such unit rotations are called binary rotations. Note that the equivalent rotations (−φ, −n) and (φ, n) are given by the same quaternion. Note also that the full range of all normalized quaternions (−1 < qi ≤ +1) includes all possible rotations twice. The inverse of a normalized quaternion obviously is a rotation about the same axis but in opposite direction: [q0 , Q]−1 = [q0 , −Q]. 14 15

(15.80)

We shall use the notation [q] or [q0 , q1 , q2 , q3 ] or [q,Q] for quaternions. Euler’s theorem: “Two arbitrarily oriented orthonormal bases with common origin P can be made to coincide with one another by rotating one of them through a certain angle about an axis which is passing through P and which has the direction of the eigenvector n of the rotation matrix” (Wittenburg, 1977).

15.7 Rigid body motion

415

r´ r´

φ

φ φ /2

r u

nx(nx r)

nx r n

r

φ/2

Figure 15.3 Rotation of the vector r to r by a rotation over an angle φ about an axis n.

We now seek the relation between quaternions and the rotation matrix or the Euler angles. The latter is rather simple if we realize that the rotation expressed in Euler angles (ϕ, ϑ, ψ) is given by R = R(ψ, k)R(ϑ, i)R(ϕ, k) = [cos

(15.81)

1 1 1 1 1 1 2 ψ, 0, 0, sin 2 ψ] [cos 2 ϑ, sin 2 ϑ, 0, 0] [cos 2 ϕ, 0, 0, sin 2 ϕ]

= [q0 , q1 , q2 , q3 ]

(15.82)

with q0 = cos 12 ϑ cos 12 (ϕ + ψ),

(15.83)

sin 12 ϑ cos 12 (ϕ − ψ), sin 12 ϑ cos 12 (ϕ − ψ), cos 12 ϑ cos 12 (ϕ + ψ).

(15.84)

q1 = q2 = q3 =

(15.85) (15.86)

The relation with the rotation matrix itself can be read from the transformation of an arbitrary vector r to r by a general rotation over an angle φ about an axis n, given by the quaternion (15.79). We must be careful with the deﬁnition of rotation: here we consider a vector to be rotated in a ﬁxed coordinate system, while R deﬁnes a transformation of the components of a ﬁxed vector to a rotated system of coordinate axes. Each rotation is the transpose of the other, and the present vector rotation is given by r = RT r. Refer to Fig. 15.3. Deﬁne a vector u, which can be constructed from the two perpendicular vectors n × (n × r) and n × r: + , u = sin 12 φ (n × (n × r)) sin 12 φ + (n × r) cos 12 φ .

416

Lagrangian and Hamiltonian mechanics

With r = r + 2u = RT r the rotation matrix can be written out and we obtain ⎞ ⎛ 2 2(q1 q2 + q0 q3 ) 2(q1 q3 − q0 q2 ) q0 + q12 − q22 − q32 R = ⎝ 2(q1 q2 − q0 q3 ) q02 − q12 + q22 − q32 2(q2 q3 + q0 q1 ) ⎠ . 2 2(q3 q2 − q0 q1 ) q0 − q12 − q22 + q32 2(q3 q1 + q0 q2 ) (15.87) Note Two remarks will be made. The ﬁrst concerns the reverse relation, from R to [q], and the second concerns the symmetry of [q] and its use in generating random rotations. The reverse relation is determined by the fact that R has the eigenvalues 1, exp(iφ) and exp(−iφ), and that the eigenvector belonging to the eigenvalue 1 is n. So, once these have been determined, [q] follows from (15.79). As can be seen from (15.87), the four quantities in [q] play an almost, but not quite symmetric role: q0 diﬀers from q1 , q2 , q3 . This is important if we would fancy to generate a random rotation. The correct procedure, using quaternions, would be to generate a vector n randomly distributed over the unit sphere, then generate a random angle φ in the range (0, 2π), and construct the quaternion from (15.79). The vector n could be obtained16 by choosing three random numbers x, y, z from a homogeneous distribution between −1 and +1, computing r2 = x2 + y 2 + z 2 , discarding all triples for which r2 > 1, and scaling all accepted triples to unit length by division by r. Such a procedure cannot be used for the four dimensions of [q], simply because q0 is diﬀerent, e.g., for a random 3D rotation q02 = 1/2 while q12 = q22 = q32 = 1/6. To generate a small random rotation in a given restricted range, as might be required in a Monte Carlo procedure, ﬁrst generate a random n, then choose a value of φ in the range Δφ as required (both (0, Δφ) and (−Δφ/2, Δφ/2) will do), and construct [q] from (15.79).

What we ﬁnally need is the rate of change of [q] due to the angular velocity ω. Noting that [q] rotates vectors rather than coordinate axes, so that [q] ˙ is T ˙ equal to R , which is given by (15.60): ˙ T = RT Ω , R

(15.88)

we can cast this into a quaternion multiplication [q] ˙ = [q] [0, 12 ω ].

(15.89)

Here we have used the fact that Ω is the time derivative of an angular rotation around an axis in the direction of ω; the factor 1/2 comes from the derivative of sin 12 φ. After working this out (reader, please check!) we ﬁnd 16

See Allen and Tildesley (1987), Appendix G4.

15.8 Holonomic constraints

the time derivative of [q]: ⎛ ⎞ ⎞ ⎛ ⎛ ⎞ −q1 −q2 −q3 q˙0 ⎜ q˙1 ⎟ 1 ⎜ q0 −q3 q2 ⎟ ωx ⎜ ⎟ ⎟ ⎝ ωy ⎠ . ⎜ ⎝ q˙2 ⎠ = 2 ⎝ q3 q0 −q1 ⎠ ωz q˙3 −q2 q1 q0

417

(15.90)

In simulations numerical and integration errors may produce a slow drift in the constraint q02 + q12 + q22 + q32 = 1, which is usually compensated by regular scaling of the q’s.

15.8 Holonomic constraints Holonomic constraints depend only on coordinates and can be described by a constraint equation σ(r) = 0 that should be satisﬁed at all times. For every constraint there is such an equation. Examples are (we use the notation r ij = r i − r j ): • distance constraint between two particles: |r 12 |−d12 = 0, or, alternatively, (r 12 )2 − d212 = 0; • angle 1-2-3 constraint between two constrained bonds: r 12 · r 32 − c = 0, where c = d12 d32 cos φ, or, alternatively, r 213 − d213 = 0. The way to introduce holonomic constraints into the equations of motion is by minimizing the action while preserving the constraints, using Lagrange multipliers. Thus we add each of the m constraints σs (q), s = 1, . . . , m with their undetermined Lagrange multipliers λs to the Lagrangian and minimize the action. As can be easily veriﬁed, this results in the modiﬁed Lagrange equations (compare (15.2)) ∂L d ∂L − =0 i = 1, . . . , n, (15.91) dt ∂ q˙k ∂qk where L = L + def

m

λs σs (q),

(15.92)

σs (q) = 0, s = 1, . . . , m

(15.93)

s=1

while for all q along the path

Equations (15.91) and (15.93) fully determine the path, i.e., both q(t) and λ(t). In fact the path is restricted to a hypersurface determined by the constraint equations. There are n+m variables (the n q’s and the m λ’s) and an

418

Lagrangian and Hamiltonian mechanics

equal number of equations (n Lagrange equations and m constraint equations). Note that the generalized momenta are not modiﬁed by holonomic ˙ but the forces constraints17 because the constraints are not functions of q, are. The total generalized force is built up from an unconstrained force and a constraint force: ∂L ∂σs ∂L = + λs . ∂qk ∂qk ∂qk m

(15.94)

s=1

If the constraints are not eliminated by the use of generalized coordinates, the λ’s must be solved from the constraint equations. We can distinguish two ways to obtain the solution, both of which will be worked out in more detail in the following subsections. The ﬁrst method, which is historically the oldest and in practice the most popular one, was devised by Ryckaert et al. (1977). It resets the coordinates after an unconstrained time step, so as to satisfy the constraints to within a given numerical precision, and therefore prevents the propagation of errors. The method is most suitable in conjunction with integration algorithms that do not contain explicit velocities, although a velocity variant is also available (Andersen, 1983). The second method rewrites the equations of motion to include the constraints by solving the λ’s from the fact that the σs ’s are zero at all times: hence all time derivatives of σs are also zero. This allows an explicit solution for the Lagrange multipliers, but the solutions contain the velocities, and since only derivatives of the constraints appear, errors may propagate. We shall call this class of solutions projection methods because in fact the accelerations are projected onto the hypersurface of constraint. The ﬁrst algorithm using this method for molecules was published by Edberg et al. (1986); the solution was cast in more general terms by de Leeuw et al. (1990) and discussed in the context of various types of diﬀerential equations by Bekker (1996). A similar method was devised by Yoneya et al. (1994), and an eﬃcient algorithm (LINCS) was published by Hess et al. (1997). We note that matrix methods to solve for holonomic as well as non-holonomic (velocitydependent) constraints were already known in the ﬁeld of macroscopic rigid body dynamics.18

17

18

That is true for our deﬁnition of constraints; de Leeuw et al. (1990) also consider a deﬁnition based on the constancy of the time derivative of σ, in which case the generalized momenta are modiﬁed, but the ﬁnal equations are the same. See, e.g., Section 5.3 of Wittenburg (1977).

15.8 Holonomic constraints

419

15.8.1 Generalized coordinates A straightforward, but often not feasible, method to implement constraints is the use of generalized coordinates in such a way that the constraints are themselves equivalent to generalized coordinates. Suppose that we transform (r 1 . . . r N ) → (q , q ),

(15.95)

where q = q1 , . . . , qn=3N −m and q = qn+1 , . . . qn+m=3N are the free and constrained coordinates, respectively. The constrained coordinates fulﬁll the m constraint equations: σs = qn+s − cs = 0.

(15.96)

Because q = c, q˙ = 0 and the kinetic energy does not contain q˙ . The Lagrangian does also not depend on q as variables, but contains the c’s only as ﬁxed parameters.Thus the q do not ﬁgure at all in the equations of motion and the Lagrangian or Hamiltonian mechanics simply preserves the constraints. The dynamics is equivalent to that produced by the use of Lagrange multipliers. The diﬃculty with this method is that only in simple cases the equations of motion can be conveniently written in such generalized coordinates.

15.8.2 Coordinate resetting One popular method of solving the constraint equations (Ryckaert et al., 1977) can be used in conjunction with the Verlet algorithm, usually applied with Cartesian coordinates: r i (t + Δt) = 2r i (t) − r i (t − Δt) +

(Δt)2 u [F i (t) + F ci (t)], mi

(15.97)

where F u are the forces disregarding the constraints, and the constraint force on particle i at time t is given by F ci (t) =

λs (t)

s

∂σs . ∂r i

(15.98)

The eﬀect of the constraint force is to add a second contribution to the displacement of the particles. The algorithm ﬁrst computes the new positions r i disregarding the constraints: r i = 2r i (t) − r i (t − Δt) +

(Δt)2 u F i (t), mi

(15.99)

420

Lagrangian and Hamiltonian mechanics Δ rj Δ ri r’i

rj (t + Δ t)

ri (t + Δ t )

r’j

d d

ri ( t )

r

tj ( )

Figure 15.4 Coordinate resetting to realize a bond length constraint.

and then corrects the positions with Δr i such that σs (r + (Δr) = 0, s = 1, . . . , m,

(15.100)

(Δt)2 ∂σs (r(t)) λs (t) . mi ∂r i s

(15.101)

where Δr i =

These equations represent a set of m (generally non-linear) coupled equations for the m λ’s, which can be solved in several ways, but as a result of the nonlinear character always requiring iteration. They can be either linearized and then solved as a set of linear equations, or the constraints can be solved sequentially and the whole procedure iterated to convergence. The latter method is easy to implement and is used in the routine SHAKE (Ryckaert et al., 1977). Let us illustrate how one distance constraint between particles i and j: σ = r 2ij − d2 = 0

(15.102)

will be reset in a partial iteration step of SHAKE. See Fig. 15.4. The particle positions are ﬁrst displaced to r . Because ∂σ/∂r i = −∂σ/∂r j = 2r ij , the displacements must be in the direction of r ij (t) and proportional to the inverse mass of each particle: 2(Δt)2 λ r ij (t), mi 2(Δt)2 λr ij (t). Δr j = − mj Δr i =

(15.103)

The variable λ is determined such that the distance between r i + Δr i and r j + Δr j is equal to d. This procedure is repeated for all constraints until all constraints have converged within a given tolerance. It is illustrative to consider the motion of a particle that is constrained to

15.8 Holonomic constraints

421

r(t−Δt)

r(t)

r(t+Δt)

r´

Figure 15.5 The action of SHAKE for a rotating particle constrained to move on a circle.

move on a circle by a distance constraint to a ﬁxed origin (Fig. 15.5). There is no external force acting on the particle, so it should move with constant angular velocity. The positions at times t − Δt and t are given. According to (15.99), the position is ﬁrst linearly extrapolated to r . Subsequently the position is reset in the direction of the constraint r(t) until the constraint is satisﬁed. Thus r(t + Δt) is obtained. It is easily seen that this algorithm gives exact results up to an angular displacement per step of close to 90◦ (four steps per period), beyond which the algorithm is unstable and fails to ﬁnd a solution. It is also seen that resetting in the direction of the constraint at time t is correct.

15.8.3 Projection methods Consider a cartesian system of point masses with equations of motion mi r¨i = F ui +

nc ∂σs s=1

∂r i

, i = 1, . . . , N,

(15.104)

which we shall write in matrix notation as M¨ x = f + CT λ,

(15.105)

where we use x for the 3N × 1 column matrix (x1 , y1 , z1 , x2 , . . . , yN , zN )T , similarly f for F u , M is the 3N × 3N diagonal matrix of masses, and the constraint matrix C is deﬁned by Csi =

∂σs . ∂xi

(15.106)

422

Lagrangian and Hamiltonian mechanics

By taking the time derivative of ˙ s=0 σ˙ s = (Cx)

(15.107)

the following relation is found: ˙ x. ˙ C¨ x = −C

(15.108)

By left-multiplying (15.105) ﬁrst by M−1 and then by C, and substituting (15.108), λ can be solved and we obtain ˙ x). ˙ λ = −(CM−1 CT )−1 (CM−1 f + C

(15.109)

The matrix CM−1 CT is non-singular and can be inverted if the constraints are independent.19 Substituting (15.109) into (15.105) we obtain the following equation of motion for x: ˙ x, ¨ = (1 − TC)M−1 f − TC ˙ x

(15.110)

T = M−1 CT(CM−1 CT )−1 .

(15.111)

where def

The matrix 1 − TC projects the accelerations due to the unconstrained forces onto the constraint hypersurface. The ﬁrst term in (15.110) gives the constrained accelerations due to the systematic forces (derivatives of the potential) and the second term gives the constrained accelerations due to centripetal forces. Equation (15.110) contains the velocities at the same time as when the forces are evaluated, but these velocities are not known at that time. Therefore this equation is in principle implicit and needs an iterative solution. In practice this is not done and not necessary. Hess et al. (1997) show that the velocities at the previous half step can be used in conjunction with appropriate corrections. The corrections are made in such a way that a stable algorithm results without drift. Although the SHAKE algorithm of Ryckaert et al. (1977) is easier to implement, the LINCS (LINear Constraint Solver) algorithm of Hess et al. (1997) is faster, more robust, more accurate and more suitable for parallel computers. For special cases it can be advantageous to solve the equations analytically, as in the SETTLE algorithm of Miyamoto and Kollman (1992) for water molecules.

19

De Leeuw et al. (1990) show that the matrix is not only non-singular but also positive deﬁnite.

16 Review of thermodynamics

16.1 Introduction and history This book is not a textbook on thermodynamics or statistical mechanics. The reason to incorporate these topics nevertheless is to establish a common frame of reference for the readers of this book, including a common nomenclature and notation. For details, commentaries, proofs and discussions, the reader is referred to any of the numerous textbooks on these topics. Thermodynamics describes the macroscopic behavior of systems in equilibrium, in terms of macroscopic measurable quantities that do not refer at all to atomic details. Statistical mechanics links the thermodynamic quantities to appropriate averages over atomic details, thus establishing the ultimate coarse-graining approach. Both theories have something to say about non-equilibrium systems as well. The logical exposition of the link between atomic and macroscopic behavior would be in the sequence: (i) (ii) (iii) (iv)

describe atomic behavior on a quantum-mechanical basis; simplify to classical behavior where possible; apply statistical mechanics to average over details; for systems in equilibrium: derive thermodynamic; quantities and phase behavior; for non-equilibrium systems: derive macroscopic rate processes and transport properties.

The historical development has followed a quite diﬀerent sequence. Equilibrium thermodynamics was developed around the middle of the nineteenth century, with the deﬁnition of entropy as a state function by Clausius forming the crucial step to completion of the theory. No detailed knowledge of atomic interactions existed at the time and hence no connection between atomic interactions and macroscopic behavior (the realm of statistical mechanics) could be made. Neither was such knowledge needed to deﬁne the state functions and their relations. 423

424

Review of thermodynamics

Thermodynamics describes equilibrium systems. Entropy is really only deﬁned in equilibrium. Still, thermodynamics is most useful when processes are considered, as phase changes and chemical reactions. But equilibrium implies reversibility of processes; processes that involve changes of state or constitution cannot take place in equilibrium, unless they are inﬁnitely slow. Processes that take place at a ﬁnite rate always involve some degree of irreversibility, about which traditional thermodynamics has nothing to say. Still, the second law of thermodynamics makes a qualitative statement about the direction of processes: a system will spontaneously evolve in the direction of increasing excess entropy (i.e., entropy in excess of the reversible exchange with the environment). This is sometimes formulated as: the entropy of the universe (= system plus its environment) can only increase. Such a statement cannot be made without a deﬁnition of entropy in a non-equilibrium system, which is naturally not provided by equilibrium thermodynamics! The second law is therefore not precise within the bounds of thermodynamics proper; the notion of entropy of a non-equilibrium system rests on the assumption that the total system can be considered as the sum of smaller systems that are locally in equilibrium. The smaller systems must still contain a macroscopic number of particles. This somewhat uneasy situation, given the practical importance of the second law, gave rise to deeper consideration of irreversible processes in the thirties and later. The crucial contributions came from Onsager (1931a, 1931b) who considered the behavior of systems that deviate slightly from equilibrium and in which irreversible ﬂuxes occur proportional to the deviation from equilibrium. In fact, the thermodynamics of irreversible processes, treating the linear regime, was born. It was more fully developed in the ﬁfties by Prigogine (1961) and others.1 In the mean time, and extending into the sixties, also the statistical mechanics of irreversible processes had been worked out, and relations were established between transport coeﬃcients (in the linear regime of irreversible processes) and ﬂuctuations occurring in equilibrium. Seminal contributions came from Kubo and Zwanzig. Systems that deviate from equilibrium beyond the linear regime have been studied extensively in the second half of the twentieth century, notably by the Brussels school of Prigogine. Such systems present new challenges: different quasi-stationary regimes can emerge with structured behavior (in time and/or in space), or with chaotic behavior. Transitions between regimes often involve bifurcation points with non-deterministic behavior. A whole new line of theoretical development has taken place since and is still active, 1

see, for example, de Groot and Mazur (1962)

16.2 Deﬁnitions

425

including chaos theory, complexity theory, and the study of emergent behavior and self-organization in complex systems. In biology, studies of this type, linking system behavior to detailed pathways and genetic make-up, are making headway under the term systems biology. In the rest of this chapter we shall summarize equilibrium thermodynamics based on Clausius’ entropy deﬁnition, without referring to the statistical interpretation of entropy. This is the traditional thermodynamics, which is an established part of both physics and chemistry. We emphasize the thermodynamic quantities related to molecular components in mixtures, traditionally treated more extensively in a chemical context. Then in Section 16.10 we review the non-equilibrium extensions of thermodynamics in the linear regime. Time-dependent linear response theory is deferred to another chapter (18). Chapter 17 (statistical mechanics) starts with the principles of quantum statistics, where entropy is given a statistical meaning.

16.2 Deﬁnitions We consider systems in equilibrium. It suﬃces to deﬁne equilibrium as the situation prevailing after the system has been allowed to relax under constant external conditions for a long time t, in the limit t → ∞. Processes that occur so slowly that in all intermediate states the system can be assumed to be in equilibrium are called reversible. State functions are properties of the system that depend only on the state of the system and not on its history. The state of a system is determined by a description of its composition, usually in terms of the number of moles2 ni of each constituent chemical species i, plus two other independent state variables, e.g., volume and temperature. State functions are extensive if they are proportional to the size of the system (such as ni ); intensive properties are independent of system size (such as the concentration of the ith species ci = ni /V ). An important intensive state function is the pressure p, which is homogeneous and isotropic in a ﬂuid in equilibrium and which can be deﬁned by the force acting per unit area on the wall of the system. Another important intensive thermodynamic state function is the temperature, which is also homogeneous in the system and can be measured by the pressure of an ideal gas in a (small, with respect to the system) rigid container in thermal contact with the system. Since a state function (say, f ) depends on the independent variables (say, x and y) only and not on processes in the past, the diﬀerential of f is a total 2

Physicists sometimes express the quantity of each constituent in mass units, but that turns out to be very inconvenient when chemical reactions are considered.

426

Review of thermodynamics

or exact diﬀerential df =

∂f ∂f dx + dy. ∂x ∂y

(16.1)

The line integral over a path from point A to point B does not depend on the path, and the integral over any closed path is zero: B df = f (B) − f (A), (16.2) A * df = 0. (16.3) If the second derivatives are continuous, then the order of diﬀerentiation does not matter: ∂2f ∂2f = . (16.4) ∂x∂y ∂y∂x As we shall see, this equality leads to several relations between thermodynamics variables. A thermodynamic system may exchange heat dq, work dw (mechanical work as well as electrical energy) and/or particles dni with its environment. We shall adopt the sign convention that dq, dw and dni are positive if heat is absorbed by the system, work is exerted on the system or particles enter the system. Both dq and dw increase the internal energy of the system. If the work is due to a volume change, it follows that dw = −p dV.

(16.5)

Neither dq nor dw is an exact diﬀerential: it is possible to extract net heat or work over a closed reversible path. We see, however, that −1/p is an integrating factor of dw, yielding the exact diﬀerential dV of a state function V . Similarly it can be shown that a function β exists, such that βdq is an exact diﬀerential. Thus the function β is an integrating factor of dq. It can be identiﬁed with the inverse absolute temperature, so that a state function S exists with dq dS = . (16.6) T The function S is called the entropy; it is an extensive state function. The entropy is only deﬁned up to a constant and the unit for S depends on the unit agreed for T . The zero point choice for the entropy is of no consequence for any process and is usually taken as the value at T = 0. In Table 16.1 the important state functions are summarized, with their S.I. units.

16.2 Deﬁnitions

427

Table 16.1 Thermodynamic state functions and their deﬁnitions and units. All variables with upper case symbols are extensive and all variables with lower case symbols are intensive, with the following exceptions: n and m are extensive; T and Mi are intensive. Deﬁnition V p T n m ρ U S H A G

a b

see text see text m/V see text see text U + pV U − TS A + pV = H − TS

ni Mi xi ci mi

ni /n ni /V ni /ms

CV Cp cV cp cV cp α κT κS μJT

(∂U/∂T )V,ni (∂H/∂T )p,ni CV /n Cp /n CV /m Cp /m (1/V )(∂V /∂T )p,ni −(1/V )(∂V /∂p)T,ni −(1/V )(∂V /∂p)S,ni (∂T /∂p)H,ni

Name

S.I. unit

volume pressure temperature total amount of moles total mass of system densitya internal energy entropy enthalpy Helmholtz free energy Gibbs free energy or Gibbs function

m3 Pa K kg kg/m3 J J/K J J J

moles of ith component molar mass of ith component mole fraction of ith component concentrationb of ith componenet molality of ith componenet (mol solute per kg solvent)

mol kg/mol mol/m3 mol/kg = molal

isochoric heat capacity isobaric heat capacity molar isochoric heat capacity molar isobaric heat capacity isochoric speciﬁc heat isobaric speciﬁc heat volume expansion coeﬀ. isothermal compressibility adiabatic compressibility Joule–Thomson coeﬃcient

J/K J/K J mol−1 K−1 J mol−1 K−1 J kg−1 K−1 J kg−1 K−1 K−1 Pa−1 Pa−1 K/Pa

The symbol ρ is sometimes also used for molar density or concentration, or for number density: particles per m3 . The unit “molar”, symbol M, for mol/dm3 = 1000 mol/m3 is usual in chemistry.

16.2.1 Partial molar quantities In Table 16.1 derivatives with respect to composition have not been included. The partial derivative yi of an extensive state function Y (p, T, ni ), with respect to the number of moles of each component, is called the partial

428

Review of thermodynamics

molar Y :

yi =

∂Y ∂ni

.

(16.7)

p,T,nj=i

For example, if Y = G, we obtain the partial molar Gibbs free energy, which is usually called the thermodynamic potential or the chemical potential : ∂G μi = , (16.8) ∂ni p,T,nj=i and with the volume V we obtain the partial molar volume vi . Without further speciﬁcation partial molar quantities are deﬁned at constant pressure and temperature, but any other variables may be speciﬁed. For simplicity of notation we shall from now on implicitly assume the condition nj =i = constant in derivatives with respect to ni . If we enlarge the whole system (under constant p, T ) by dn, keeping the mole fractions of all components the same (i.e., dni = xi dn), then the system enlarges by a fraction dn/n and all extensive quantities Y will enlarge by a fraction dn/n as well: Y (16.9) dY = dn. n But also: ∂Y dY = dni = yi xi dn. (16.10) ∂ni p,T,nj=i i

i

Hence Y =n

xi yi =

i

n i yi .

(16.11)

i

Note that this equality is only valid if the other independent variables are intensive state functions (as p and T ) and not for, e.g., V and T . The most important application is Y = G: ni μi . (16.12) G= i

This has a remarkable consequence: since μi dni + ni dμi , dG = i

(16.13)

i

but also, as a result of (16.8), (dG)p,T =

i

μi dni ,

(16.14)

16.3 Thermodynamic equilibrium relations

it follows that

ni (dμi )p,T = 0.

429

(16.15)

i

This equation is the Gibbs–Duhem relation, which is most conveniently expressed in the form xi (dμi )p,T = 0. (16.16) i

The Gibbs–Duhem relation implies that not all chemical potentials in a mixture are independent. For example, consider a solution of component s (solute) in solvent w (water). If the mole fraction of the solute is x, then xw = (1 − x) and xs = x,

(16.17)

x dμs + (1 − x) dμw = 0.

(16.18)

and

This relation allows derivation of the concentration dependence of μs from the concentration dependence of μw . The latter may be determined from the osmotic pressure as a function of concentration. There are numerous other partial molar quantities. The most important ones are ∂V vi = , (16.19) ∂ni p,T ∂U , (16.20) ui = ∂ni p,T ∂H , (16.21) hi = ∂ni p,T ∂S , (16.22) si = ∂ni p,T which are related by μi = hi − T si

and hi = ui + p vi .

(16.23)

16.3 Thermodynamic equilibrium relations The ﬁrst law of thermodynamics is the conservation of energy. If the number of particles and the composition does not change, the change in internal energy dU is due to absorbed heat dq and to work exerted on the system dw. In equilibrium, the former equals T dS and the latter −p dV , when

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Review of thermodynamics

other types of work as electrical work, nuclear reactions and radiation are disregarded. Hence dU = T dS − p dV.

(16.24)

With the deﬁnitions given in Table 16.1 and (16.8), we arrive at the following diﬀerential relations: dU = T dS − p dV + μi dni , (16.25) i

dH = T dS + V dp +

μi dni ,

(16.26)

i

dA = −S dT − p dV +

μi dni ,

(16.27)

μi dni .

(16.28)

i

dG = −S dT + V dp +

i

Each of the diﬀerentials on the left-hand side are total diﬀerentials, 12 partial diﬀerentials such as (from (16.28)): ∂G = −S, ∂T p,ni ∂G = V, ∂p T,ni ∂G = μi . ∂ni p,T

deﬁning

(16.29) (16.30) (16.31)

These are among the most important thermodynamic relations. The reader is invited to write down the other nine equations of this type. Note that the entropy follows from the temperature dependence of G. However, one can also use the temperature dependence of G/T (e.g. from an equilibrium constant), to obtain the enthalpy rather than the entropy: ∂(G/T ) = H. ∂(1/T )

(16.32)

This is the very useful Gibbs–Helmholtz relation. Being total diﬀerentials, the second derivatives of mixed type do not depend on the sequence of diﬀerentiation. For example, from (16.28): ∂2G ∂2G = , ∂p ∂T ∂T ∂p

(16.33)

16.3 Thermodynamic equilibrium relations

implies that

−

∂S ∂p

=

T,ni

∂V ∂T

431

.

(16.34)

p,ni

This is one of the Maxwell relations. The reader is invited to write down the other 11 Maxwell relations of this type.

16.3.1 Relations between partial diﬀerentials The equations given above, and the diﬀerential relations that follow from them, are a selection of the possible thermodynamic relations. They may not include a required derivative. For example, what is the relation between CV and Cp or between κS and κT ? Instead of listing all possible relations, it is much more eﬀective and concise to list the basic mathematical relations from which such relations follow. The three basic rules are given below. Relations between partial diﬀerentials f is a diﬀerentiable function of two variables. There are three variables x, y, z, which are related to each other and all partial diﬀerentials of the type (∂x/∂y)z exist. Then the following rules apply: Rule 1:

∂f ∂x

= z

∂f ∂x

+ y

∂f ∂y

x

∂y ∂x

.

(16.35)

z

Rule 2:

Rule 3:

∂x ∂y

∂x ∂y

z

= z

∂y ∂z

x

∂y ∂x

∂z ∂x

−1 (inversion).

(16.36)

= −1 (cyclic chain rule).

(16.37)

z

y

From these rules several relations can be derived. For example, in order to relate general dependencies on volume with those on pressure, one can apply Rule 1: ∂f ∂f α ∂f = + , (16.38) ∂T V ∂T p κT ∂p T or Rule 3:

∂f ∂V

T

1 =− κT V

∂f ∂p

. T

(16.39)

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Review of thermodynamics

Useful relations are α2 V T , κT α2 V T κT = κS + , Cp ∂p ∂U p = T − . ∂T V ∂V T

Cp = CV +

(16.40) (16.41) (16.42)

Equation (16.42) splits pressure into an ideal gas kinetic part and an internal part due to internal interactions. The term (∂U/∂V )T indicates deviation from ideal gas behavior. The proofs are left to the exercises at the end of this chapter.

16.4 The second law Thus far we have used the second law of thermodynamics in the form dS = dq/T , valid for systems in equilibrium. The full second law, however, states that dq dS ≥ (16.43) T for any system, including non-equilibrium states. It tells us in what direction spontaneous processes will take place. When the system has reached full equilibrium, the equality holds. This qualitative law can be formulated for closed systems for three diﬀerent cases: • Closed, adiabatic system: when neither material nor heat is exchanged with the environment (dq = 0), the system will spontaneously evolve in the direction of maximum entropy: dS ≥ 0.

(16.44)

In equilibrium S is a maximum. • Closed, isothermal and isochoric system: when volume and temperature are kept constant (dV = 0, dT = 0), then dq = dU and T dS ≥ dU . This implies that dA ≤ 0.

(16.45)

The system will spontaneously evolve in the direction of lowest Helmholtz free energy. In equilibrium A is a minimum.

16.5 Phase behavior

433

• Closed, isothermal and isobaric system: When pressure and temperature are kept constant (dp = 0, dT = 0), then dq = dH and T dS ≥ dH. This implies that dG ≤ 0.

(16.46)

The system will spontaneously involve in the direction of lowest Gibbs free energy. In equilibrium G is a minimum. For open systems under constant p and T that are able to exchange material, we can formulate the second law as follows: the system will spontaneously evolve such that the thermodynamic potential of each component becomes homogeneous. Since G = i ni μi , the total G would decrease if particles would move from a region where their thermodynamic potential is high to a region where it is lower. Therefore particles would spontaneously move until their μ would be the same everywhere. One consequence of this is that the thermodynamic potential of any component is the same in two (or more) coexisting phases. 16.5 Phase behavior A closed system with a one-component homogeneous phase (containing n moles) has two independent variables or degrees of freedom, e.g., p and T . All other state functions, including V , are now determined. Hence there is relation between p, V, T : Φ(p, V, T ) = 0,

(16.47)

which is called the equation of state (EOS). Examples are the ideal gas EOS: p v = RT (where v is the molar volume V /n), or the van der Waals gas (p + a/v 2 )(v − b) = RT . If two phases, as liquid and gas, coexist, there is the additional restriction that the thermodynamic potential must be equal in both phases, and only one degree of freedom (either p or T ) is left. Thus there is a relation between p and T along the phase boundary; for the liquid– vapor boundary boiling point and vapor pressure are related. When three phases coexist, as solid, liquid and gas, there is yet another restriction which leaves no degrees of freedom. Thus the triple point has a ﬁxed temperature and pressure. Any additional component in a mixture adds another degree of freedom, viz. the mole fraction of the additional component. The number of degrees of freedom F is related to the number of components C and the number of coexisting phases P by Gibbs’ phase rule: F = C − P + 2,

(16.48)

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Review of thermodynamics C

xC

xB A

xA

(a)

B

(b)

Figure 16.1 (a) points representing mole fractions of 3 components A,B,C in a cartesian coordinate system end up in the shaded triangle (b) Each vertex represents a pure component and each mole fraction is on a linear 0-1 scale starting at the opposite side. The dot represents the composition xA = 0.3, xB = 0.5, xC = 0.2.

as the reader can easily verify. A phase diagram depicts the phases and phase boundaries. For a single component with two degrees of freedom, a two-dimensional plot suﬃces, and one may choose as independent variables any pair of p, T , and either V or molar density ρ = n/V . Temperature–density phase diagrams contain a coexistence region where a single phase does not exist and where two phases (gas and liquid, or solid and liquid) are in equilibrium, one with low density and one with high density. In simulations on a small system a densitytemperature combination in the coexistence region may still yield a stable ﬂuid with negative pressure. A large amount of real ﬂuid would separate because the total free energy would then be lower, but the separation is a slow process, and in a small system the free energy cost to create a phase boundary (surface pressure) counteracts the separation. For mixtures the composition comes in as extra degrees of freedom. Phase diagrams of binary mixtures are often depicted as x, T diagrams. Ternary mixtures have two independent mole fractions; if each of the three mole fractions are plotted along three axes of a 3D cartesian coordinate system, the condition i xi = 1 implies that all possible mixtures lie on the plane through the points (1,0,0), (0,1,0) and (0,0,1) (Fig. 16.1a). Thus any composition can be depicted in the corresponding triangle (Fig. 16.1b). Along a phase boundary between two phases 1 and 2 in the T, p plane we know that the thermodynamic potential at every point is equal on both sides of the boundary. Hence, stepping dT, dp along the boundary, dμ1 = dμ2 : dμ1 = v1 dp − s1 dT = dμ2 = v2 dp − s2 dT.

(16.49)

16.6 Activities and standard states

435

Therefore, along the boundary the following relation holds Δs 1 Δh dp = = , (16.50) dT Δv T Δv where Δ indicates a diﬀerence between the two phases. These relations are exact. If we consider the boiling or sublimation line, and one phase can be approximated by an ideal gas and the other condensed phase has a negligible volume compared to the gas, we may set Δv = RT /p, and we arrive at the Clausius–Clapeyron equation for the temperature dependence of the saturation pressure Δhvap d ln p = . (16.51) dT RT 2 This equation implies that the saturation pressure increases with temperature as −Δhvap . (16.52) p(T ) ∝ exp RT 16.6 Activities and standard states The thermodynamic potential for a gas at low pressure, which approaches the ideal gas for which the molar volume v = RT /p, is given by p p 0 0 μ(p) = μ (p ) + v dp = μ0 (p0 ) + RT ln 0 . (16.53) p p0 Here μ0 is the standard thermodynamic potential at some standard pressure p0 . For real gases at non-zero pressure the thermodynamic potential does not follow this dependence exactly. One writes γp μ(p) = μ0 + RT ln 0 , (16.54) p where γ is the activity coeﬃcient and f = γp is called the fugacity of the gas. It is the pressure the gas would have had it been ideal. For p → 0, γ → 1. For solutions the thermodynamic potential of a dilute solute behaves in a similar way. When the concentration (or the mole fraction or molality) of a component approaches zero, the thermodynamic potential of that component becomes linear in the logarithm of the concentration (mole fraction, molality): μ(c) = μ0c + RT ln(γc c/c0 ), μ(m) = μ(x) =

μ0m + RT ln(γm m/m0 ), μ0x + RT ln(γx x).

(16.55) (16.56) (16.57)

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Review of thermodynamics

The standard concentration c0 is usually 1 M (molar = mol dm−3 ), and the standard molality m0 is 1 mole per kg solvent. For mole fraction x the standard reference is the pure substance, x = 1. The γ’s are activity coeﬃcients and the products γc c/c0 , γm m/m0 , γx x are called activities; one should clearly distinguish these three diﬀerent kinds of activities. They are, of course, related through the densities and molar masses. Note that μ(c0 ) = μ0c and μ(m0 ) = μ0m , unless the activity coeﬃcients happen to be zero at the standard concentration or molality. The deﬁnition of μ0 is c def μ0c = lim μ(c) − RT ln 0 , (16.58) c→0 c and similarly for molalities and for mole fractions of solutes. For mole fractions of solvents the standard state x = 1 represents the pure solvent, and μ0x is now deﬁned as μ(x = 1), which is usually indicated by μ∗ . For x = 1 the activity coeﬃcient equals 1. Solutions that have γx = 1 for any composition are called ideal. The reader is warned about the inaccurate use, or incomplete deﬁnitions, of standard states and activities in the literature. Standard entropies and free energies of transfer from gas phase to solution require proper deﬁnition of the standard states in both phases. It is very common to see ln c in equations, meaning ln(c/c0 ). The logarithm of a concentration is mathematically undeﬁned.

16.6.1 Virial expansion For dilute gases the deviation from ideal behavior can be expressed in the virial expansion, i.e., the expansion of p/RT in the molar density ρ = n/V : p = ρ + B2 (T )ρ2 + B3 (T )ρ3 + · · · . kB T

(16.59)

This is in fact an equation of state for the dilute gas phase. The second virial coeﬃcient B2 is expressed in m3 /mol. It is temperature dependent with usually negative values at low temperatures and tending towards a limiting positive value at high temperature. The second virial coeﬃcient can be calculated on the basis of pair interactions and is therefore an important experimental quantity against which an interaction function can be calibrated.3 For dilute solutions a similar expansion can be made of the 3

See Hirschfelder et al. (1954) for many details on determination and computation of virial coeﬃcients.

16.7 Reaction equilibria

437

osmotic pressure (see (16.85)) versus the concentration: Π = c + B2 (T )c2 + B3 (T )c3 + · · · . kB T

(16.60)

The activity coeﬃcient is related to the virial coeﬃcients: using the expression 1 ∂p ∂μ = , (16.61) ∂ρ T ρ ∂ρ T we ﬁnd that 3 (16.62) μ(ρ, T ) = μideal + 2RT B2 (T )ρ + RT B3 (T )ρ2 + · · · . 2 This implies that 3 (16.63) ln γc = 2B2 ρ + B3 ρ2 + · · · . 2 Similar expressions apply to the fugacity and the osmotic coeﬃcient.

16.7 Reaction equilibria Consider reaction equilibria like A + 2 B AB2 , which is a special case of the general reaction equilibrium 0 νi Ci

(16.64)

i

Here, Ci are the components, and νi the stoichiometric coeﬃcients, positive on the right-hand side and negative on the left-hand side of the reaction. For the example above, νA = −1, νB = −2 and νAB2 = +1. In equilibrium (under constant temperature and pressure) the total change in Gibbs free energy must be zero, because otherwise the reaction would still proceed in the direction of decreasing G: νi μi = 0. (16.65) i

Now we can write μi = μ0i + RT ln ai ,

(16.66)

where we can ﬁll in any consistent standard state and activity deﬁnition we desire. Hence νi μ0i = −RT νi ln ai . (16.67) i

i

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Review of thermodynamics

The left-hand side is a thermodynamic property of the combined reactants, usually indicated by ΔG0 of the reaction, and the right-hand side can also be expressed in terms of the equilibrium constant K: def ΔG0 = νi μ0i = −RT ln K, (16.68) i def

K = Πi aνi i .

(16.69)

The equilibrium constant depends obviously on the deﬁnitions used for the activities and standard states. In dilute solutions concentrations or molalities are often used instead of activities; note that such equilibrium “constants” are not constant if activity coeﬃcients deviate from 1. Dimerization 2 A A2 in a gas diminishes the number of “active” particles and therefore reduces the pressure. In a solution similarly the osmotic pressure is reduced. This leads to a negative second virial coeﬃcient B2 (T ) = −Kp RT for dilute gases (Kp = pA2 /p2A being the dimerization constant on the basis of pressures) and B2 (T ) = −Kc for dilute solutions (Kc = cA2 /c2A being the dimerization constant on the basis of concentrations). A special case of an equilibrium constant is Henry’s constant, being the ratio between pressure in the gas phase of a substance A, and mole fraction xA in dilute solution.4 It is an inverse solubility measure. The reaction is A(sol, x) A(gas, p) with standard state x = 1 in solution and p = p0 in the gas phase. Henry’s constant KH = p/x relates to the standard Gibbs free energy change as ΔG0 = μ0 (gas; p0 ) − μ0 (sol; x = 1) = −RT ln KH .

(16.70)

Other special cases are equilibrium constants for acid–base reactions involving proton transfer, and for reduction–oxidation reactions involving electron transfer, both of which will be detailed below.

16.7.1 Proton transfer reactions The general proton transfer reaction is HA H+ + A− , where the proton donor or acid HA may also be a charged ion (like NH+ 4 + or HCO− 3 ) and H stands for any form in which the proton may appear in 4

The inverse of KH , expressed not as mole fraction but as molality per bar, is often tabulated (e.g., by NIST). It is also called Henry’s law constant and denoted by kH .

16.7 Reaction equilibria

439

solution (in aqueous solutions most likely as H3 O+ ). A− is the corresponding proton acceptor or base (like NH3 or (CO3 )2− ). The equilibrium constant in terms of activities based on molar concentrations is the acid dissociation constant Ka : a +a − [H+ ][A− ] , (16.71) Ka = H A ≈ aHA [HA]c0 where the brackets denote concentrations in molar, and c0 = 1 M.5 When the acid is the solvent, as in the dissociation reaction of water itself: H2 O H+ + OH− , the standard state is mole fraction x = 1 and the dissociation constant Kw = 10−14 is simply the product of ionic concentrations in molar. With the two deﬁnitions def

(16.72)

def

(16.73)

pH = − log10 aH+ ≈ − log10 [H+ ] pKa = − log10 Ka , we ﬁnd that ΔG0 = −RT ln Ka = 2.3026 RT pKa .

(16.74)

It is easily seen that the acid is halfway dissociated (activities of acid and base are equal) when the pH equals pKa . 16.7.2 Electron transfer reactions The general electron transfer reaction involves two molecules (or ions): a donor D and an acceptor A: D + A D+ + A−

.

In this process the electron donor is the reductant that gets oxidized and the electron acceptor is the oxidant that gets reduced. Such reactions can be formally built up from two separate half-reactions, both written as a reduction: D+ + e− D A + e− A−

, .

The second minus the ﬁrst reaction yields the overall electron transfer reaction. Since the free electron in solution is not a measurable intermediate,6 5 6

Usually, c0 is not included in the deﬁnition of K, endowing K with a dimension (mol dm−3 ), and causing a formal inconsistency when the logarithm of K is needed. Solvated electrons do exist; they can be produced by radiation or electron bombardment. They have a high energy and a high reductive potential. In donor-acceptor reactions electrons are transferred through contact, through material electron transfer paths or through vacuum over very small distances, but not via solvated electrons.

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Review of thermodynamics

one cannot attach a meaning to the absolute value of the chemical potential of the electron, and consequently to the equilibrium constant or the ΔG0 of half-reactions. However, in practice all reactions involve the diﬀerence between two half-reactions and any measurable thermodynamic quantities involve diﬀerences in the chemical potential of the electron. Therefore such quantities as μe and ΔG0 are still meaningful if a proper reference state is deﬁned. The same problem arises if one wishes to split the potential diﬀerence of an electrochemical cell (between two metallic electrodes) into two contributions of each electrode. Although one may consider the potential diﬀerence between two electrodes as the diﬀerence between the potential of each electrode with respect to the solution, there is no way to measure the “potential of the solution.” Any measurement would involve an electrode again. The required standard is internationally agreed as the potential of the standard hydrogen electrode, deﬁned as zero with respect to the solution (at any temperature). The standard hydrogen electrode is a platinum electrode in a solution with pH = 0 and in contact with gaseous hydrogen at a pressure of 1 bar. The electrode reduction half-reaction is 2 H+ + 2 e− H2 As the electron is in equilibrium with the electrons in a metallic electrode at a given electrical potential Φ, under conditions of zero current, the thermodynamic potential of the electron is given by μe = −F Φ,

(16.75)

where F is the Faraday, which is the absolute value of the charge of a mole electrons (96 485 C). 1 Volt corresponds to almost 100 kJ/mol. Here, the electrical potential is deﬁned with respect to the “potential of the solution” according to the standard hydrogen electrode convention. We can now summarize the thermodynamic electrochemical relations for the general half-reaction ox + ν e− red as follows: ΔG = μred − μox − νμe = 0,

(16.76)

μ0red + RT ln ared − μ0ox − RT ln aox + νF Φ = 0.

(16.77)

implying that

16.8 Colligative properties

441

With the deﬁnition of the standard reduction potential E 0 : −νF E 0 = ΔG0 = μ0red − μ0ox ,

(16.78)

we arrive at an expression for the equilibrium, i.e., current-free, potential7 of a (platinum) electrode with respect to the “potential of the solution” (deﬁned through the standard hydrogen electrode) Φ = E0 −

ared RT ln . νF aox

(16.79)

Values of E 0 have been tabulated for a variety of reduction–oxidation couples, including redox couples of biological importance. When a metal or other solid is involved, the activity is meant with respect to mole fraction, which is equal to 1. The convention to tabulate reduction potentials is now universally accepted, meaning that a couple with a more negative standard potential has – depending on concentrations – the tendency to reduce a couple with a more positive E 0 . A concentration ratio of 10 corresponds to almost 60 mV for a single electron transfer reaction. 16.8 Colligative properties Colligative properties of solutions are properties that relate to the combined inﬂuence of all solutes on the thermodynamic potential of the solvent. This causes the solvent to have an osmotic pressure against the pure solvent and to show shifts in vapor pressure, melting point and boiling point. For dilute solutions the thermodynamic potential of the solvent is given by ∗

∗

μ = μ + RT ln xsolv = μ + RT ln(1 −

xj )

(16.80)

j

≈ μ∗ − RT

j

xj ≈ μ∗ − Msolv RT

mj ,

(16.81)

j

μ∗

is the thermodynamic potential of the pure solvent at the same where pressure and temperature, and the prime in the sum means omitting the solvent itself. Msolv is the molar mass of the solvent. A solution with a thermodynamic potential of the solvent equal to an ideal dilute solution of m molal has an osmolality of m. The consequences of a reduced thermodynamic potential of the solvent ∗ μ − Δμ are the following: 7

The current-free equilibrium potential has long been called the electromotive force (EMF), but this historical and confusing nomenclature is now obsolete. The inconsistent notation E (usual for electric ﬁeld, not potential) for the standard potential has persisted, however.

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Review of thermodynamics

(i) The vapor pressure p is reduced with respect to the saturation vapor pressure p∗ of the pure solvent (assuming ideal gas behavior) according to p (16.82) −Δμ = RT ln ∗ , p or ∗ xj ). (16.83) p = p (1 − j

This is a form of Raoult’s law stating that the vapor pressure of a volatile component in an ideal mixture is proportional to the mole fraction of that component. (ii) The solution has an osmotic pressure Π (to be realized as a real pressure increase after the pure solvent is equilibrated with the solution, from which it is separated by a semipermeable membrane that is exclusively permeable to the solvent), determined by the equality μ∗ = μ∗ − Δμ + Πvsolv , or Π = ρsolv RT mosmol ≈

RT

(16.84)

j

vsolv

xj

.

(16.85)

The latter equation is the van ’t Hoﬀ equation stating that the osmotic pressure equals the pressure that the solute would have if the solute particles would behave as in an ideal gas. (iii) The solution has a boiling point elevation ΔTb equal to Δμ , sg − sl

(16.86)

Δμ ΔTb = . Tb Δhvap

(16.87)

ΔTb = or

In Fig. 16.2 the boiling point elevation is graphically shown to be related to the reduction in thermodynamic potential of the solvent. Since the slopes of the lines are equal to the molar entropies, equation (16.86) follows directly from this plot, under the assumption that the curvature in the range (Tb0 , Tb ) is negligible, (iv) Similarly, the solution has a freezing point depression (or melting point depression) ΔTm (see Fig. 16.2), given by ΔTm =

Δμ , sl − ss

(16.88)

16.9 Tabulated thermodynamic quantities

443

Thermodynamic potential μ solid pure solvent solution

vapor Tm Tmo

Tbo Tb Temperature

Figure 16.2 Thermodynamic potential of the solvent for solid, liquid and vapor phases, both for the pure liquid (thick lines) and for a solution (thin line), as a function of temperature. The negative slope of the curves is given by the molar entropy of each state. The state with lowest μ is stable. The solution shows a melting point depression and boiling point elevation, proportional to the reduction of the solvent thermodynamic potential.

or Δμ ΔTm = . Tm Δhfusion

(16.89)

16.9 Tabulated thermodynamic quantities Thermodynamic quantities, derived from experiments, need unambiguous speciﬁcation when tabulated for general use. This applies to standard states and activities and to standard conditions, if applicable. Standard states should include the quantity and the unit in which the quantity is expressed. In the case of pressure, the now preferred standard state is 1 bar (105 Pa), but the reader should be aware that some tables still use the standard atmosphere, which equals 101 325 Pa. In all cases the temperature and pressure, and other relevant environmental parameters, should be speciﬁed. Good tables include error estimates, give full references and are publicly accessible.

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Review of thermodynamics

Quantities of formation (usually enthalpy, entropy and Gibbs free energy) of a substance refer to the formation of the substance from its elements, all components taken at the reference temperature of 273.15 K and reference pressure of 1 bar, in the phase in which the substance or element is stable under those conditions (if not speciﬁed otherwise). The absolute entropy refers to the third-law entropy, setting S = 0 at T = 0. The international organization CODATA maintains a list of key values for thermodynamics.8 Other sources for data are the Handbook of Chemistry and Physics,9 and various databases10 of which many are freely available through the (US) National Institute of Standards and Technology (NIST).11

16.10 Thermodynamics of irreversible processes Consider a system that is slightly oﬀ-equilibrium because small gradients exist of temperature, pressure, concentrations, and/or electric potential. The gradients are so small that the system may be considered locally in equilibrium over volumes small enough to have negligible gradients and still large enough to contain a macroscopic number of particles for which thermodynamic relations apply. We now look at two adjacent compartments (see Fig. 16.3), each in local equilibrium, but with slightly diﬀerent values of parameters such as temperature, pressure, concentration and/or electric potential. There are possibly ﬂuxes from the left to the right compartment or vice-versa, both of heat and of particles.

16.10.1 Irreversible entropy production The crucial step is to consider the total entropy production in both compartments together. We shall then see that the ﬂuxes cause an entropy production. This irreversible entropy production, which is always positive, can be related to the product of properly deﬁned ﬂuxes and thermodynamic forces which are the gradients that cause the ﬂuxes. Start with (16.25). Since the volume of the compartments is constant, 8

9 10 11

CODATA, the Committee on Data for Science and Technology, is based in Paris and acts under the auspices of the International Council for Science ICSU,. See http://www.codata.org. For the key values see http://www.codata.org/codata/databases/key1.html. Published by CRC Press, Baco Raton, FL, USA. It comes out in a yearly printed edition and in a Web version available at http://www.hbcpnetbase.com A useful list of links to data sites is provided by http://tigger.uic.edu/∼mansoori/Thermodynamic.Data.and.Property.html. http://www.nist gov.

16.10 Thermodynamics of irreversible processes

1

445

2 T + ΔT

T

Ji

p c

p + Δp

Jq

Φ

Δx

x

Δy Δ z

c + Δc Φ + ΔΦ

x + Δx

Figure 16.3 Two adjacent compartments in a system in which irreversible ﬂuxes of particles and/or heat are present. Each compartment of size ΔxΔyΔz is in local equilibrium, but adjacent compartments have slightly diﬀerent values of temperature, pressure, concentrations and/or electric potential representing gradients in the x-direction.

(16.25) can be rewritten as dS =

dU μi − dni . T T

(16.90)

i

Numbering the compartments 1 and 2, the total entropy change equals dS1 + dS2 =

μ2i dU1 dU2 μ1i + − dn1 − dn2 . T1 T2 T1 T2 i

(16.91)

i

Now call the entropy production per unit volume and per unit time σ. If we call the energy ﬂux per unit time and per unit of surface area Ju (in J m−2 s−1 ) and the particle ﬂux Ji (in mol m−2 s−1 ), then dU1 dU2 =− , dt dt dn1i dn2i =− , Ji Δy Δz = dt dt

Ju Δy Δz =

so that

σ Δx Δy Δz = Ju

1 1 − T2 T1

−

i

Ji

μ2i μ1i − T2 T1

(16.92) (16.93) Δy Δz.

(16.94)

Equating the diﬀerences between 1 and 2 to the gradient multiplied by Δx,

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Review of thermodynamics

and then extending to three dimensions, we ﬁnd μ 1 i − σ = Ju · ∇ Ji · ∇ . T T

(16.95)

i

This equation formulates the irreversible entropy production per unit volume and per unit time as a sum of scalar products of ﬂuxes J α and conjugated forces X α : J α · X α = JT X, (16.96) σ= α

where the gradient of the inverse temperature is conjugate to the ﬂux of internal energy, and minus the gradient of the thermodynamic potential divided by the temperature, is conjugate to the particle ﬂux. The last form in (16.96) is a matrix notation, with J and X being column vector representations of all ﬂuxes and forces. It is possible and meaningful to transform both ﬂuxes and forces such that (16.96) still holds. The formulation of (16.94) is very inconvenient: for example, a temperature gradient does not only work through the ﬁrst term but also through the second term, including the temperature dependence of the thermodynamic potential. Consider as independent variables: p, T, Φ, xi , i = 1, . . . , n − 112 and use the following derivatives: ∂μi = vi , ∂p

∂μi = −si , ∂T

∂μi = zi F, ∂Φ

(16.97)

where zi is the charge (including sign) in units of the elementary charge of species i. Equation (16.94) then transforms to 1 1 1 1 − J v · ∇p + j · E − J i · (∇μi )p,T . (16.98) σ = Jq · ∇ T T T T i

Here we have used the relation μi = hi − T si and the following deﬁnitions: • J q is the heat ﬂux (in J m−2 s−1 ), being the energy ﬂux from which the contribution as a result of particle transport has been subtracted: def J i hi . (16.99) Jq = Ju − i

Note that the energy transported by particles is the partial molar enthalpy, not internal energy, because there is also work performed against the pressure when the partial molar volume changes. 12

With n components there are n − 1 independent mole fractions. One may also choose n − 1 concentrations or molalities.

16.10 Thermodynamics of irreversible processes

447

• J v is the total volume ﬂux (in m/s), which is a complicated way to express the bulk velocity of the material: def J i vi . (16.100) Jv = i

• j is the electric current density (in A/m2 ): def J i zi F. j =

(16.101)

i

Note that the irreversible entropy production due to an electric current is the Joule heat divided by the temperature. The last term in (16.98) is related to gradients of composition and needs to be worked out. If we have n species in the mixture, there are only n − 1 independent composition variables. It is convenient to number the species with i = 0, 1, . . . , n − 1, with i = 0 representing the solvent, and xi , i = 1, . . . , n − 1 the independent variables. Then x0 = 1 −

xi ,

(16.102)

i

where the prime in the summation means omission of i = 0. The Gibbs– Duhem relation (16.16) relates the changes in chemical potential of the different species. It can be written in the form x0 (∇μ0 )p,T +

xi (∇μi )p,T = 0.

(16.103)

i

Using this relation, the last term in (16.98) can be rewritten as

1 xi 1 J i − J 0 · (∇μi )p,T . J i · (∇μi )p,T = − − T T x0 i

(16.104)

i

Here a diﬀerence ﬂux with respect to ﬂow of the solvent appears in the equation for irreversible entropy production. If all particles move together at the same bulk ﬂux J (total moles per m2 and per s), then Ji = xi J for all i, and Jid = 0. So this deﬁnition makes sense: a concentration gradient produces a diﬀerence ﬂux by irreversible diﬀusion processes. Note that relation (16.98), including (16.104), has the form of a product of ﬂuxes and forces as in (16.96). Hence also for these ﬂuxes and forces linear and symmetric Onsager relations (Section 16.10.3) are expected to be valid.

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16.10.2 Chemical reactions Whenever chemical reactions proceed spontaneously, there is also an irreversible entropy production. According to (16.90), entropy is generated when the composition changes according to −(1/T ) i μi dni . In the previous subsection this term was evaluated when the number of molecules changed due to ﬂuxes. We have not yet considered what happens if the numbers change due to a chemical reaction. Assume the reaction13 0→ νi Ci (16.105) i

proceeds for a small amount, Δn mole, as written. This means that νi Δn moles of species Ci are formed (or removed when νi is negative). The irreversible entropy production ΔS is 1 ΔS = − ( νi μi )Δn. T

(16.106)

i

With the following deﬁnitions: def

• the aﬃnity of the reaction A = −

i νi μi ; def

• the degree of advancement of the reaction ξ = number of moles per unit volume the reaction (as written) has proceeded, the irreversible entropy production per unit volume and per unit time due to the advancement of the reaction can be written as σ=

1 dξ A . T dt

(16.107)

The rate of advancement dξ/dt can be viewed as the reaction ﬂux and A/T then is the driving force for the reaction. Note that reaction ﬂuxes and forces are scalar quantities, contrary to the vector quantities we encountered thus far. The rate of advancement is equal to the usual net velocity of the reaction vreact =

1 d[Ci ] . νi dt

(16.108)

In equilibrium, the aﬃnity is zero. For reactions that deviate only slightly from equilibrium, the velocity is linear in the aﬃnity; far from equilibrium no such relation with aﬃnity exists. 13

We use the same notation as in (16.64).

16.10 Thermodynamics of irreversible processes

449

16.10.3 Phenomenological and Onsager relations For small deviations from equilibrium, i.e., small values of the driving forces X, one may assume that the ﬂuxes are proportional to the driving forces. Such linear relations are the ﬁrst term in a Taylor expansion in the driving forces,14 and they are justiﬁed on a phenomenological basis. The main driving force for a ﬂux is its conjugated force, but in general the linear relations may involve any other forces as well: Lkl Xl or J = LX. (16.109) Jk = l

Here Lkl are the phenomenological coeﬃcients. This implies that for the entropy production σ = XT LT X.

(16.110)

From the second law we know that the irreversible entropy production must always be positive for any combination of driving forces. Mathematically this means that the matrix L must be positive deﬁnite with only positive eigenvalues. The diagonal phenomenological coeﬃcients relate to ﬂuxes resulting from their conjugate forces, such as heat conduction (heat ﬂow due to temperature gradient), viscous ﬂow, e.g., through a membrane or through porous material (ﬂuid ﬂow due to hydrostatic pressure diﬀerence), electrical conduction (current due to electric ﬁeld), and diﬀusion (particle ﬂow with respect to solvent due to concentration gradient). Oﬀ-diagonal coeﬃcients relate to ﬂuxes that result from other than their conjugate forces, such as: • thermoelectric eﬀect (current due to temperature gradient) and Peltier eﬀect (heat ﬂow due to electric ﬁeld); • thermal diﬀusion or Soret eﬀect (particle separation due to temperature gradient) and Dufour eﬀect (heat ﬂow due to concentration gradient); • osmosis (volume ﬂow due to concentration gradient) and reverse osmosis (particle separation due to pressure gradient); • electro-osmosis (volume ﬂow due to electric ﬁeld) and streaming potential (current due to pressure gradient); • diﬀusion potential (current due to concentration gradient) and electrophoresis (particle separation due to electric ﬁeld). On the basis of microscopic reversibility and by relating the phenomenological coeﬃcients to ﬂuctuations, Onsager (1931b) came to the conclusion 14

Note that this does not imply that the coeﬃcients themselves are constants; they may depend on temperature, pressure and concentration.

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that the matrix L must be symmetric: Lkl = Llk .

(16.111)

These are Onager’s reciprocal relations that relate cross eﬀects in pairs. For example, the thermoelectric eﬀect is related to the Peltier eﬀect as follows: 1 E + Lqe , (16.112) T T 1 E (16.113) j = Leq grad + Lee . T T The thermoelectric coeﬃcient is deﬁned as the ratio of the potential difference, arising under current-free conditions, to the externally maintained temperature diﬀerence (in V/K). This is equal to Leq grad Φ =− . (16.114) grad T j=0 T Lee Jq = Lqq grad

The Peltier eﬀect is deﬁned as the heat ﬂux carried per unit current density under conditions of constant temperature (in J/C). This is equal to Lqe Jq =− . (16.115) j grad T =0 Lee Onsager’s relation Leq = Lqe implies that the thermoelectric coeﬃcient equals the Peltier coeﬃcient, divided by the absolute temperature.

16.10.4 Stationary states When there are no constraints on a system, it will evolve spontaneously into an equilibrium state, in which the irreversible entropy production becomes zero. The direction of this process is dictated by the second law; a ﬂow Ji diminishes the conjugate force, and in course of time the entropy production decreases. With external constraints, either on forces or on ﬂuxes, or on combinations thereof, the system will evolve into a stationary state (or steady state), in which the entropy production becomes minimal. This is easily seen as follows. Assume that forces Xα are constrained by the environment. Xα are a subset of all forces Xi . The system develops in the direction of decreasing entropy production, until the entropy production is minimal. Minimizing XT LX under constraints Xk = constant, requires that ∂ ( Lij Xi Xj + λα Xα ) = 0, (16.116) ∂Xk α ij

Exercises

451

for all k, and where λα are Lagrange undetermined multipliers. This implies that: • Jα = constant; • Jk = 0 for k = α. Thus, the system evolves into a steady state with constant ﬂuxes; ﬂuxes conjugate to unconstrained forces vanish.

Exercises 16.1

Show that the following concentration dependencies, valid for ideal solutions, satisfy the Gibbs–Duhem relation: μw = μ0w + RT ln(1 − x), μs = μ0s + RT ln x.

16.2 16.3 16.4 16.5 16.6 16.7

16.8

16.9

Prove the Gibbs–Helmholtz equation, (16.32). Show that another Gibbs–Helmholtz equation exists, replacing G by A and H by U . At ﬁrst sight you may wonder why the term i μi dni occurs in (16.25) to (16.28). Prove, for example, that (∂H/∂ni )p,T = μi . Prove (16.40) and (16.41). Prove (16.42) by starting from the pressure deﬁnition from (16.27), and then using the Maxwell relation derived from (16.27). Estimate the boiling point of water from the Clausius–Clapeyron equation at an elevation of 5500 m, where the pressure is 0.5 bar. The heat of vaporization is 40 kJ/mol. Estimate the melting point of ice under a pressure of 1 kbar. The densities of ice and water are 917 and 1000 kg/m3 ; the heat of fusion is 6 kJ/mol. Rationalize that the cryoscopic constant (the ratio between freezing point depression and molality of the solution) equals 1.86 K kg mol−1 for water.

17 Review of statistical mechanics

17.1 Introduction Equilibrium statistical mechanics was developed shortly after the introduction of thermodynamic entropy by Clausius, with Boltzmann and Gibbs as the main innovators near the end of the nineteenth century. The concepts of atoms and molecules already existed but there was no notion of quantum theory. The link to thermodynamics was properly made, including the interpretation of entropy in terms of probability distributions over ensembles of particle conﬁgurations, but the quantitative counting of the number of possibilities required an unknown elementary volume in phase space that could only later be identiﬁed with Planck’s constant h. The indistinguishability of particles of the same kind, which had to be introduced in order to avoid the Gibbs’ paradox,1 got a ﬁrm logical basis only after the invention of quantum theory. The observed distribution of black-body radiation could not be explained by statistical mechanics of the time; discrepancies of this kind have been catalysts for the development of quantum mechanics in the beginning of the twentieth century. Finally, only after the completion of basic quantum mechanics around 1930 could quantum statistical mechanics – in principle – make the proper link between microscopic properties at the atomic level and macroscopic thermodynamics. The classical statistical mechanics of Gibbs is an approximation to quantum statistics. In this review we shall reverse history and start with quantum statistics, proceeding to classical statistical mechanics as an approximation to quantum 1

In a conﬁguration of N particles there are N ! ways to order the particles. If each of these ways are counted as separate realizations of the conﬁguration, a puzzling paradox arises: thermodynamic quantities that involve entropy appear not to be proportional to the size of the system. This paradox does not arise if the number of realizations is divided by N !. In quantum statistics one does not count conﬁgurations, but quantum eigenstates which incorporate the indistinguishability of particles of the same kind in a natural way.

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statistics. This will enable us to see the limitations of classical computational approaches and develop appropriate quantum corrections where necessary. Consider an equilibrium system of particles, like nuclei and electrons, possibly already pre-organized in a set of molecules (atoms, ions). Suppose that we know all the rules by which the particles interact, i.e., the Hamiltonian describing the interactions. Then we can proceed to solve the time-independent Schr¨ odinger equation to obtain a set of wave functions and corresponding energies, or – by classical approximation – a set of conﬁgurations in phase space, i.e., the multidimensional space of coordinates and momenta with their corresponding energies. The aim of statistical mechanics is to provide a recipe for the proper averaging over these detailed solutions in order to obtain thermodynamic quantities. Hopefully the averaging is such that the (very diﬃcult and often impossible) computation of the detailed quantities can be avoided. We must be prepared to handle thermodynamic quantities such as temperature and entropy which cannot be obtained as simple averages over microscopic variables.

17.2 Ensembles and the postulates of statistical mechanics The basic idea, originating from Gibbs,2 is to consider a hypothetical ensemble of a large number of replicas of the system, with the same thermodynamic conditions but diﬀerent microscopic details. Properties are then obtained by averaging over the ensemble, taken in the limit of an inﬁnite number of replicates. The ensemble is supposed to contain all possible states of the system and be representative for the single system considered over a long time. This latter assumption is the ergodic postulate. Whether a realistic system is in practice ergodic (i.e., are all microscopic possibilities indeed realized in the course of time?) is a matter of time scale: often at low temperatures internal processes may become so slow that not all possibilities are realized in the experimental observation time, and the system is not ergodic and in fact not in complete equilibrium. Examples are metastable crystal modiﬁcations, glassy states, polymer condensates and computer simulations that provide incomplete sampling or insuﬃcient time scales. Let us try to set up an appropriate ensemble. Suppose that we can describe discrete states of the system, numbered by i = 1, 2, . . . with energies Ei . In quantum mechanics, these states are the solutions of the Schr¨ odinger 2

Josiah Willard Gibbs (1839–1903) studied mathematics and engineering in Yale, Paris, Berlin and Heidelberg, and was professor at Yale University. His major work on statistical mechanics dates from 1876 and later; his collected works are available (Gibbs, 1957). See also http://wwwgap.dcs.st-and.ac.uk/∼history/Mathematicians/Gibbs.html.

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455

equation for the whole system.3 Note that energy levels may be degenerate, i.e., many states can have the same energy; i numbers the states and not the distinct energy levels. In classical mechanics a state may be a point in phase space, discretized by subdividing phase space into elementary volumes. Now envisage an ensemble of N replicas. Let Ni = wi N copies be in state i with energy Ei . Under the ergodicity assumption, the fraction wi is the probability that the (single) system is to be found in state i. Note that wi = 1. (17.1) i

The number of ways W the ensemble can be made up with the restriction of given Ni ’s equals N! (17.2) W= Πi Ni ! because we can order the systems in N ! ways, but should not count permutations among Ni as distinct. Using the Stirling approximation for the factorial,4 we ﬁnd that ln W = −N wi ln wi . (17.3) i

If the assumption is made (and this is the second postulate of statistical mechanics) that all possible ways to realize the ensemble are equally probable, the set of probabilities {wi } that maximizes W is the most probable distribution. It can be shown that in the limit of large N , the number of ways the most probable distribution can be realized approaches the total number of ways that any distribution can be realized, i.e., the most probable distribution dominates all others.5 Therefore our task is to ﬁnd the set of probabilities {wi } that maximizes the function H=− wi ln wi . (17.4) i

This function is equivalent to Shannon’s deﬁnition of information or uncertainty over a discrete probability distribution (Shannon, 1948).6 It is 3 4 5 6

The states as deﬁned here are microscopic states unrelated to the thermodynamic states deﬁned in Chapter 16. √ N ! ≈ N N e−N 2πN {1 + (O)(1/N )}. For the present application and with N → ∞, the approximation N ! ≈ N N e−N suﬃces. More precisely: the logarithm of the number of realizations of the maximum distribution approaches the logarithm of the number of all realizations in the limit of N → ∞. The relation with information theory has led Jaynes (1957a, 1957b) to propose a new foundation for statistical mechanics: from the viewpoint of an observer the most unbiased guess he can make about the distribution {wi } is the one that maximizes the uncertainty H under the constraints of whatever knowledge we have about the system. Any other distribution would

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also closely related (but with opposite sign) to the H-function deﬁned by Boltzmann for the classical distribution function for a system of particles in coordinate and velocity space. We shall see that this function is proportional to the entropy of thermodynamics.

17.2.1 Conditional maximization of H The distribution with maximal H depends on further conditions we may impose on the system and the ensemble. Several cases can be considered, but for now we shall concentrate on the N, V, T or canonical ensemble. Here the particle number and volume of the system and the expectation of the energy, i.e., the ensemble-averaged energy i wi Ei , are ﬁxed. The systems are allowed to interact weakly and exchange energy. Hence the energy per system is not constant, but the systems in the ensemble belong to the same equilibrium conditions. Thus H is maximized under the conditions wi = 1, (17.5)

i

wi Ei = U.

(17.6)

i

Using the method of Lagrange multipliers,7 the function − wj ln wj + α wj − β wj E j j

j

(17.7)

j

(the minus sign before β is for later convenience) is maximized by equating all partial derivatives to wi to zero: −1 − ln wi + α − βEi = 0,

(17.8)

wi ∝ e−βEi .

(17.9)

or

7

be a biased choice that is only justiﬁed by additional knowledge. Although this principle leads to exactly the same results as the Gibbs postulate that all realizations of the ensemble are equally probable, it introduces a subjective ﬂavor into physics that is certainly not universally embraced. Lagrange undetermined multipliers are used to ﬁnd the optimum of a function f (x) of n variables x under s constraint conditions of the form gk (x) = 0, k = 1, . . . , s. One constructs the function f + sk=1 λk gk , where λk are as yet undetermined multipliers. The optimum of this function is found by equating all partial derivatives to zero. Then the multipliers are solved from the constraint equations.

17.3 Identiﬁcation of thermodynamical variables

457

The proportionality constant (containing the multiplier α) is determined by normalization condition (17.5), yielding 1 −βEi e , Q e−βEi . Q =

wi =

(17.10) (17.11)

i

Q is called the canonical partition function. The multiplier β follows from the implicit relation 1 U= Ei e−βEi . (17.12) Q i

As we shall see next, β is related to the temperature and identiﬁed as 1/kB T .

17.3 Identiﬁcation of thermodynamical variables Consider a canonical ensemble of systems with given number of particles and ﬁxed volume. The system is completely determined by its microstates labeled i with energy Ei , and its thermodynamic state is determined by the probability distribution {wi }, which in equilibrium is given by the canonical distribution (17.10). The distribution depends on one (and only one) parameter β. We have not introduced the temperature yet, but it must be clear that the temperature is somehow related to the distribution {wi }, and hence to β. Supplying heat to the system has the consequence that the distribution {wi } will change. In the following we shall ﬁrst identify the relation between β and temperature and between the distribution {wi } and entropy. Then we shall show how the partition function relates to the Helmholtz free energy, and – through its derivatives – to all other thermodynamic functions.

17.3.1 Temperature and entropy Now consider the ensemble-averaged energy, which is equal to the thermodynamic internal energy U : wi E i . (17.13) U= i

The ensemble-averaged energy changes by changing the distribution {wi }, corresponding to heat exchange dq: Ei dwi . (17.14) dU = dq = i

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Note that, as a result of the normalization of the probabilities, dwi = 0.

(17.15)

i

At constant volume, when no work is done on the system, the internal energy can only change by absorption of heat dq, which in equilibrium equals T dS: 1 dq = dS. (17.16) dU = dq = T dS or T So, in thermodynamics the temperature is deﬁned as the inverse of the integrating factor of dq that produces the diﬀerential dS of a state function S (see the discussion on page 426). Can we ﬁnd an integrating factor for dq in terms of the probabilities wi ? From (17.10) follows that Ei = −β −1 ln wi − β −1 ln Q, which can be inserted in (17.14) to yield ln wi dwi . dq = −β −1

(17.17)

(17.18)

i

Here use has been made of the fact that i dwi = 0. Using this fact again, it follows that β dq = d(− wi ln wi ). (17.19) i

So we see that β is an integrating factor for dq, yielding a total diﬀerential of a thermodynamic state function − i wi ln wi . Therefore this state function can be identiﬁed with the entropy S and β with the inverse temperature 1/T . Both functions can be scaled with an arbitrary constant, which is determined by the convention about units in the deﬁnition of temperature. Including the proper constant we conclude that 1 , kB T wi ln wi . S = −kB β =

(17.20) (17.21)

i

These are the fundamental relations that couple statistical mechanics and thermodynamics.8 Note that the entropy is simply equal to the information function H introduced in (17.4), multiplied by Boltzmann’s constant. 8

Several textbooks use these equations as deﬁnitions for temperature and entropy, thus ignoring the beautiful foundations of classical thermodynamics.

17.4 Other ensembles

459

Strictly, the entropy is only deﬁned by (17.21) in the case that {wi } represents a canonical equilibrium distribution. We may, however, extend the deﬁnition of entropy by (17.21) for any distribution; in that case ﬁnding the equilibrium distribution is equivalent to maximizing the entropy under the constraint that i wi = 1 and the additional constraints given by the deﬁnition of the ensemble (for the canonical ensemble: constant N and V and given expectation for the energy U : U = i wi Ei ). 17.3.2 Free energy and other thermodynamic variables The entropy is proportional to the expectation of ln wi , i.e., the average of ln wi over the distribution {wi }: S = −kB ln wi .

(17.22)

From the canonical distribution (17.10), it follows that ln wi = − ln Q − βEi ,

(17.23)

and taking the expectation over both sides, we obtain −

S U , = − ln Q − kB kB T

(17.24)

which reshuﬄes to −kB T ln Q = U − T S = A.

(17.25)

This simple relation between Q and the Helmholtz free energy A is all we need to connect statistical and thermodynamic quantities: if we know Q as a function of V and β, we know A as a function of V and T , from which all other thermodynamic quantities follow.

17.4 Other ensembles Thus far we have considered the canonical ensemble with constant N and V , and given expectation U of the energy over the ensemble. It appeared that the latter requirement implied the existence of a constant β, identiﬁed with the inverse temperature. Thus the canonical ensemble is also called the N, V, T ensemble. Although the canonical ensemble is often the most useful one, it is by no means the only possible ensemble. For example, we can constrain not only N and V , but also the energy E for each system and obtain the microcanonical ensemble; the ensemble then consists of systems in diﬀerent microstates

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(wave functions) with the same degenerate energy.9 Instead of constraining the volume for each system, we can prescribe the ensemble average of the volume, which introduces another constant that appears to be related to the pressure. This is the N, p, T ensemble. Finally, if we do not ﬁx the particle number for each system, but only ﬁx the ensemble average of the particle number, a constant appears that is related to the chemical potential. This produces the grand-canonical or μ, V, T ensemble if the volume is ﬁxed, or the μ, p, T ensemble if the ensemble-averaged volume is ﬁxed. The recipe is always the same: Let wi be the probability that the system is in state i (numbering each of all possible states, given the freedom we give the various parameters), and maximize − i wi ln wi under the conditions we impose on the ensemble. Each condition introduces one Lagrange multiplier, which can be identiﬁed with a thermodynamic quantity. This can be summarized as follows: • The N, V, E or microcanonical ensemble. The system will have a degeneracy Ω, being the number of states with energy E (or within a very small, ﬁxed energy interval). − i wi ln wi is maximized under the only condi tion that i wi = 1, which implies that all probabilities wi are equal and equal to 1/Ω; it follows that S = kB ln Ω. Knowledge of the “partition function” Ω (and hence S) as a function of V and E = U then generates all thermodynamic quantities. For example, T = ∂S/∂E. • The N, V, T or canonical ensemble. See above. The partition function is Q(N, V, β) = i exp(−βEi ) and thermodynamic functions follow from β = (kB T )−1 and A = −kB T ln Q. • The N, p, T or isobaric-isothermal ensemble. Here the particle number of the system and the ensemble-averaged energy and volume are ﬁxed. The wi are now a function of volume (a continuous variable), and we look for the probability wi (V ) dV that the system is in state i with volume between V and V +dV . Thus10 H = − dV i wi (V ) ln wi (V ) is maximized under the conditions dV wi (V ) = 1, (17.26) i 9

10

Neither experimentally, nor in simulations is it possible to generate an exact microcanonical ensemble. The spacing between energy levels becomes very small for macroscopic systems and complete thermal isolation including radiation exchange is virtually impossible; algorithms usually do not conserve energy exactly. But the microcanonical ensemble can be deﬁned as having an energy in a small interval (E, E + ΔE). This is a somewhat sloppy extension of the H-function of (17.4) with an integral. The Hfunction becomes inﬁnite when a continuous variable is introduced, because the number of choices in the continuous variable is inﬁnite. The way out is to discretize the variable V in small intervals ΔV . The equation for H then contains ln[wi (V ) ΔV ]. But for maximization the introduction of ΔV is immaterial.

17.4 Other ensembles

dV

wi (V )Ei (V ) = U,

i

dV V

461

(17.27)

wi (V ) = V.

(17.28)

ln wi (V ) ∝ −βEi (V ) − γV,

(17.29)

i

The Lagrange optimization yields

or 1 −βEi (V )−γV e , Δ e−βEi (V )−γV Δ = dV

wi (V ) =

(17.30) (17.31)

i

Δ is the isothermal-isobaric partition function. Identifying −kB ln wi (V ) with the entropy S, we ﬁnd that S = ln Δ + βU + γV. kB

(17.32)

Hence, β = (kB T )−1 , γ = βp, and the thermodynamic functions follow from G = U − T S + pV = −kB T ln Δ.

(17.33)

• The μ, V, T or grand-canonical ensemble.11 Here the ensemble averages of E and N are ﬁxed, and a Lagrange multiplier δ is introduced, related to the condition N = NA n, where NA is Avogadro’s number and n is the (average) number of moles in the system.12 The microstates now involve every possible particle number N and all quantum states for every N . The probabilities and partition function are then given by 1 −βEN ,i +δN e , Ξ eδN e−βEN ,i . Ξ =

wN,i =

N

(17.34) (17.35)

i

Working out the expression for the entropy S = −kB ln wN,i and comparing with the thermodynamic relation T S = U + pV − nμ, 11 12

(17.36)

Often just called “the grand ensemble.” For a multi-component system, there is a given average number of particles and a Lagrange multiplier for each species. Many textbooks do not introduce Avogadro’s number here, with the consequence that the thermodynamic potential is deﬁned per particle and not per mole as is usual in chemistry.

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one ﬁnds the identiﬁcations β = (kB T )−1 , δ = μ/RT and pV = kB T ln Ξ(μ, V, T ).

(17.37)

This equation relates the grand partition function to thermodynamics. Note that ∞ μN . (17.38) QN exp Ξ= RT N =0

If we deﬁne the absolute activity λ as:13 μ def λ = exp( , RT then the grand partition function can be written as Ξ=

∞

λN QN .

(17.39)

(17.40)

N =0

Partial derivatives yield further thermodynamic functions: λ

∂ ln Ξ = N = NA n, ∂λ p ∂ ln Ξ = , ∂V kB T ∂ ln Ξ = −U. ∂β

(17.41) (17.42) (17.43)

This ends our review of the most important ensembles. In simulations one strives for realization of one of these ensembles, although integration errors and deviations from pure Hamiltonian behavior may cause distributions that are not exactly equal to those of a pure ensemble. If that is the case, one may in general still trust observed averages, but observed ﬂuctuations may deviate signiﬁcantly from those predicted by theory.

17.4.1 Ensemble and size dependency One may wonder if and if so, why, the diﬀerent ensembles yield the same thermodynamic functions. After all, the various ensembles diﬀer in the freedom that we allow for the system and therefore their entropies are diﬀerent as well. It is quite clear that the entropy of a given system is larger in a canonical than in a microcanonical ensemble, and larger still in a grand ensemble, because there are more microstates allowed. This would seemingly 13

The name “absolute activity” is logical if we compare μ = RT ln λ with the deﬁnition of activity a (e.g., (16.66)): μ = μ0 + RT ln a.

17.5 Fermi–Dirac, Bose–Einstein and Boltzmann statistics

463

lead to diﬀerent values for the entropy, as well as for other thermodynamic functions. The point is that, although the entropies are not strictly the same, they tend to the same value when the system is macroscopic and contains a large number of particles. Each of the thermodynamic variables that is not ﬁxed per system has a probability distribution over the ensemble that tends to a delta function in the limit of an inﬁnite system, with the same value for each kind of ensemble. The ensembles do diﬀer, however, both in the values of averages and in ﬂuctuations, for systems of ﬁnite size with a ﬁnite number of particles. In numerical simulations in particular one deals with systems of ﬁnite size, and one should be aware of (and correct for) the ﬁnite-size eﬀects of the various ensembles. Let us, just for demonstration, consider the ﬁnite-size eﬀects in a very simple example: a system of N non-interacting spins, each of which can be either “up” or “down”. In a magnetic ﬁeld the total energy will be proportional to i mi . Compare the microcanonical ensemble with exact energy E = 0, requiring 12 N spins up and 12 N spins down, with the canonical ensemble at such high temperature that all 2N possible conﬁgurations are equally likely (the Boltzmann factor for any conﬁguration equals 1). The entropy in units of kB is given by microcanonical : S = ln

N! [( 12 N )!]2

canonical : S = N ln 2.

= ln N ! − 2 ln( 12 N )!,

(17.44) (17.45)

We see that for large N , in the limit where the Stirling approximation ln N ! ≈ N ln N − N is valid, the two entropies are equal. For smaller N this is not the case, as Fig. 17.1 shows. Plotting the “observed” entropy versus N −1 ln N allows extrapolation to inﬁnite N .

17.5 Fermi–Dirac, Bose–Einstein and Boltzmann statistics In this section ﬁrst a more general formulation for the canonical partition function will be given in terms of the trace of an exponential matrix in Hilbert space. The reader may wish to review parts of Chapter 14 as an introduction. Then we shall look at a system of non-interacting particles (i.e., an ideal gas) where the symmetry properties of the total wave function appear to play a role. For fermions this leads to Fermi–Dirac (FD) statistics, while bosons obey Bose–Einstein (BE) statistics. In the limit of low density or high temperature both kinds of statistics merge into the Boltzmann approximation.

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Smicro 1000

500

300

200

100

N

0.70 2000

Scan

ln 2

0.69

0.68

0.67

0.66

0.01

0.02

0.03

0.04

0.05 1 N

ln N

Figure 17.1 Diﬀerence between canonical and microcanonical ensemble entropies for ﬁnite systems. Case: N non-interacting spins in a magnetic ﬁeld at high temperature. The canonical entropy per mole in units of R (or per spin in units of kB ) equals ln 2, independent of system size (solid line); the microcanonical (at E = 0) molar entropy depends on N and tends to ln 2 for large N . Extrapolation is nearly linear if S is plotted against N −1 ln N : the dashed line is a linear connection between the data points at N = 1000 and N = 2000.

17.5.1 Canonical partition function as trace of matrix Consider a system of N particles in a box of volume V . For simplicity we take a cubic box and assume the system to be inﬁnite and periodic with the box as repeating unit cell. These restrictions are convenient but not essential: the system may contain various types of diﬀerent particles, or have another shape. Although not rigorously proved, it is assumed that eﬀects due to particular choices of boundary conditions vanish for large system sizes since these eﬀects scale with the surface area and hence the eﬀect per particle scales as N −1/3 . The wave function Ψ(r 1 , r 2 , . . .) of the N -particle system can be constructed as a linear combination of properly (anti)symmetrized products of single-particle functions. The N -particle wave function must

17.5 Fermi–Dirac, Bose–Einstein and Boltzmann statistics

465

change sign if two identical fermions are interchanged, and remain the same if two identical bosons are interchanged (see Chapter 2, page 37, for details). In equilibrium we look for stationary quantum states involving all particles. There will be stationary solutions with wave functions Ψi and energies Ei and the canonical partition function is given by exp(−βEi ). (17.46) Q= i

Consider a Hilbert space spanned by all the stationary solutions Ψi of the Hamiltonian (see Chapter 14 for details of vectors and transformations in Hilbert spaces). Then the matrix H is diagonal and Ei = Hii . Thus we can also write Q = tr exp(−βH).

(17.47)

This equality is quite general and also valid on any complete basis set on which the Hamiltonian is not diagonal. This is easily seen by applying a unitary transformation U that diagonalizes H, so that U † HU is diagonal, and realizing that (see (14.30) on page 386) exp(−βU † HU ) = U † exp(−βH)U,

(17.48)

and that, because the trace of a product is invariant for cyclic exchange of the elements in the product, tr U † AU = tr U U † A = tr A.

(17.49)

Solving Q would now require the computation of all diagonal elements of the Hamiltonian (on a complete basis set). This seems simpler than solving the Schr¨ odinger equation for the whole system, but is still in practice impossible for all but the simplest systems.

17.5.2 Ideal gas: FD and BE distributions In order to get insight into the eﬀect of the symmetry requirements of the wave function on the partition function we shall now turn to a system which is solvable: the ideal gas. In the ideal gas there are no interactions between particles. We shall also, for convenience, but without loss of generality, assume that the system contains identical particles, which are either fermions or bosons. Let the single-particle wave functions be given by φk (r) with energy εk (Fig. 17.2). The φk form an orthonormal set of functions, and the

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.. . • •

k 6 φk , εk , nk

• • •

Figure 17.2 Single-particle quantum states k, with wave function φk , energy εk and occupation number nk . For fermions nk is restricted to 0 or 1. The shown double occupation of the third level is only possible for bosons.

total wave function is an (anti)symmetrized sum of product states 1 P [φk1 (r 1 )φk2 (r 2 ) . . .]. (17.50) Ψi (r 1 , r 2 , . . .) = √ (∓1)P N! P Here the sum is over all possible N ! permutations of the N particles, and (∓1)P is a shorthand notation for −1 in case of an odd number of permutations of two fermions (the upper sign) and +1 in case of bosons (lower sign). √ The factor 1/ N ! is needed to normalize the total wave function again. It is clear that the total wave function vanishes if two fermions occupy the same single-particle wave function φk . Therefore the number of particles nk occupying wave function k is restricted to 0 or 1, while no such restriction exists for bosons: nk = 0, 1 (fermions).,

(17.51)

nk = 0, 1, 2, . . . (bosons).

(17.52)

A N -particle wave function is characterized by the set of occupation numbers n = {n1 , n2 , . . . , nk , . . .} with the restriction N= nk . (17.53) k

The energy En is given by En =

n k εk .

(17.54)

k

All possible states with all possible numbers of particles are generated by all possible sets n of numbers subject to the condition (17.51) for fermions.

17.5 Fermi–Dirac, Bose–Einstein and Boltzmann statistics

467

Thus the grand partition function equals Ξ =

∞

λN

n

N =0

= = =

e−βEn

···λ

k

nk −β

e

k

nk εk

n1

n2

nk · · · Πk λe−βεk

n1

n2

n1

n1

= Πk

λe−βε1

λe−βεk

n2 nk

λe−βε2

n2

···

.

(17.55)

nk

where each sum over nk runs over the allowed values. For fermions only the values 0 or 1 are allowed, yielding the Fermi–Dirac statistics: ΞFD = Πk 1 + λe−βεk . (17.56) For bosons all values of nk are allowed, yielding the Bose–Einstein statistics: −1 . (17.57) ΞBE = Πk 1 + λe−βεk + λ2 e−2βεk + . . . = Πk 1 − λe−βεk Equations (17.56) and (17.57) can be combined as ±1 −βεk 1 ± λe = Π ΞFD k BE with thermodynamic relation βpV = ln Ξ = ±

ln 1 ± λe−βεk

(17.58)

(17.59)

k

and occupancy numbers given by nk =

exp[−β(εk − μ/NA )] λ exp(−βεk ) = 1 ± λ exp(−βεk ) 1 ± exp[−β(εk − μ/NA )]

(17.60)

(upper sign: FD; lower sign: BE). Figure 17.3 shows that fermions will ﬁll low-lying energy levels approximately until the thermodynamic potential per particle (which is called the Fermi level) is reached; one example is electrons in metals (see exercises). Bosons tend to accumulate on the lowest levels; the thermodynamic potential of bosons is always lower than the lowest level. This Bose condensation phenomenon is only apparent at very low temperatures.

468

Review of statistical mechanics nk 2 1.5 BE 1

B FD

0.5 0 −4

−2

0

2

4 β(εk − μ/NAv)

Figure 17.3 Occupation number nk of kth single-particle quantum state, in an ideal quantum gas, as function of the energy level εk above the thermodynamic potential μ, for fermions (FD) and bosons (BE). The dashed line indicates the classical approximation (Boltzmann statistics).

17.5.3 The Boltzmann limit In gases at high temperature or low density the number of available quantum states considerably exceeds the number of particles. In a periodic box of dimensions a × a × a; V = a3 , the functions 2π n, n ∈ Z3 , (17.61) a are eigenfunctions of the kinetic energy operator, with eigenvalues φk = V −1/2 exp(ikr),

k=

2 k 2 . (17.62) 2m This implies (see Exercise 17.3) that the number of single-particle translational quantum states between ε and ε + dε is given by εk =

√ m3/2 V 1/2 g(ε) dε = 4π 2 ε dε. h3

(17.63)

Consider 1 cm3 of neon gas at a pressure of 1 bar and temperature of 300 K, containing 2.4×1019 atoms. By integrating (17.63) from 0 up to kB T we ﬁnd that there are 6.7 × 1025 quantum states with energies up to kB T . Thus the probability that any particular quantum state is occupied is much smaller than one, and the probability that any state is doubly occupied is negligible.

17.5 Fermi–Dirac, Bose–Einstein and Boltzmann statistics

469

Therefore there will be no distinction between fermions and bosons and the system will behave as in the classical or Boltzmann limit. In this limit the occupancies nk 1 and hence λe−βεk 1.

(17.64)

In the Boltzmann limit, including the lowest order deviation from the limit, the occupancies and grand partition function are given by (upper sign: FD; lower sign: BE) nk ≈ λe−βεk ∓ λ2 e−2βεk + · · · , βpV = ln Ξ ≈ λ e−βεk ∓ 12 λ2 k e−2βεk + · · · .

(17.65) (17.66)

k

Since N = k nk , the ideal gas law pV = N kB T is recovered in the Boltzmann limit. The ﬁrst-order deviation from ideal-gas behavior can be expressed as a second virial coeﬃcient B2 (T ): p kB T NA−1 B2 (T )FD BE

2 N N + NA−1 B2 (T ) + ···, V V h3 Λ3 = ± = ± , 2(4πmkB T )3/2 25/2 ≈

(17.67) (17.68)

where Λ is the de Broglie thermal wavelength def

Λ = √

h . 2πmkB T

(17.69)

Avogadro’s number comes in because B2 is deﬁned per mole and not per molecule (see (16.59)). This equation is proved as follows: ﬁrst show that NA−1 B2 = ±q2 V /(2q 2 ), where q = k exp(−βεk ) and q2 = k exp(−2βεk ), and then solve q and q2 by approximating the sum by an integral: q=

k

e−βεk ,

εk =

h2 n2 2 k 2 = . 2m 2ma2

(17.70)

Here, n2 = n2x + n2y + n2z with nx , ny , nz ∈ {0, ±1, ±2, . . .} (see (17.61)). Use has been made of the periodicity requiring that k = (2π/a)n. Since the occupation numbers are high (n is large), the sum can be approximated by q≈

∞

−∞

∞

−∞

a3 βh2 2 V dn = exp − n (2πmkB T )3/2 = 3 . (17.71) 2 3 2ma h Λ −∞ ∞

470

Review of statistical mechanics

Note that this q is the single particle canonical translational partition function of a particle in a periodic box.14 Also note that the quantum deviation from the Boltzmann limit,15 due to particle symmetry, is of the order h3 . In the Boltzmann limit, the grand partition function Ξ and the singleparticle canonical partition function q are related by ln Ξ = λq and thus Ξ = eλq =

N

λN

(17.72) qN . N!

(17.73)

Since Ξ = N λN QN (see (17.40)), it follows that the N -particle canonical partition function QN for non-interacting particles equals qN . (17.74) N! The N ! means that any state obtained by interchanging particles should not be counted as a new microstate, as we expect from the indistinguishability of identical quantum particles. It is a result of quantum symmetry that persists in the classical limit. It’s omission would lead to thermodynamic functions that are neither intensive nor extensive (the Gibbs’ paradox) as the following will show. Consider a gas of N non-interacting atoms in a periodic box of volume V , with translational single-atom partition function (17.71) QN =

V (17.75) Λ3 Using (17.74) and the Stirling approximation (see footnote on page 455) for N !, the Helmholtz free energy is given by q=

qN A = −kB T ln QN = −kB T ln N −N N e q q − N kB T = −N kB T ln − pV. (17.76) = −N kB T ln N N From this follows the absolute thermodynamic potential of the gas μ = 14

15

A + pV q G = = −RT ln n n N

(17.77)

The same result is obtained if the box is not periodic, but closed with inﬁnitely high walls. The wave functions must than vanish at the boundaries and thus be composed of sine waves with wave lengths that are whole fractions of twice the box length. This leads to 8× higher density of points in n-space, of which only one octant (positive n) is valid, and thus to the same value of the integral. Any quantum correction to classical behavior should contain h; the classical limit is often viewed as the limit for h → 0, which is nonsensical for a physical constant, but useful.

17.5 Fermi–Dirac, Bose–Einstein and Boltzmann statistics

= RT ln

p Λ 3 p0 + RT ln 0 , kB T p

471

(17.78)

where p0 is (any) standard pressure. We recover the linear dependence of μ of the logarithm of the pressure. Without the N ! we would have found μ to be proportional to the logarithm of the pressure divided by the number of particles: a nonsensical result. The single-particle partition function q is still fully quantum-mechanical. It consists of a product of the translational partition function qtrans (computed above) and the internal partition function, which – in good approximation – consists of a product of the rotational partition function qrot and the internal vibrational partition function qvib , all for the electronic ground state. If there are low-lying excited electronic states that could be occupied at the temperatures considered, the internal partition function consists of a sum of vibro-rotational partition functions, if applicable multiplied by the degeneracy of the electronic state, for each of the relevant electronic states. The rotational partition function for a linear molecule with moment of inertia I (rotating in 2 dimension) equals

2 J(J + 1) qrot = (17.79) (2J + 1) exp − 2IkB T J

Θ 1 Θ 2 T 1+ + + ... , (17.80) = σΘ 3T 15 T def

where Θ = 2 /(2IkB ), and σ is the symmetry factor. The summation is over the symmetry-allowed values of the quantum number J: for a homonuclear diatomic molecule σ = 2 because J can be either even or odd, depending on the symmetry of the wave function on interchange of nuclei, and on the symmetry of the spin state of the nuclei. The high-temperature limit for the linear rotator, valid for most molecules at ordinary temperatures, is qrot =

2IkB T . σ2

(17.81)

For a general non-linear molecule rotating in three dimensions, with moment of inertia tensor16 I, the high-temperature limit for the partition function is given by (2kB T )3/2 π det(I). (17.82) qrot = σ3 In contrast to the rotational partition function, the vibrational partition 16

For the deﬁnition of the inertia tensor see (15.52) on page 409.

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Review of statistical mechanics

function can in general not be approximated by its classical high-temperature limit. Low-temperature molecular vibrations can be approximated by a set of independent harmonic oscillators (normal modes with frequency νi ) and the vibrational partition function is a product of the p.f. of each normal mode. A harmonic oscillator with frequency ν has equidistant energy levels (if the minimum of the potential well is taken as zero energy): εn = (n + 12 )hν,

n = 0, 1, 2, . . . ,

(17.83)

and the partition function is qho =

exp(− 12 ξ) = 12 [sinh(ξ/2)]−1 , 1 − exp(−ξ)

(17.84)

where ξ = hν/kB T . In the low-temperature limit only the ﬁrst level is occupied and q tends to exp(− 12 ξ) (or one if the lowest level is taken as zero energy); in the high-temperature (classical) limit q tends to kB T /hν. Figure 17.4 compares the quantum-statistical partition function, (free) energy and entropy with the classical limit: although the diﬀerence in Q is not large, the diﬀerence in S is dramatic. The classical entropy tends to −∞ as T → 0, which is a worrying result! For temperatures above hν/kB the classical limit is a good approximation.

17.6 The classical approximation A full quantum calculation of the partition function of a multidimensional system is in general impossible, also if the Hamiltonian can be accurately speciﬁed. But, as quantum dynamics for “heavy” particles can be approximated by classical mechanics, quantum statistics can be approximated by a classical version of statistical mechanics. In the previous section we considered the classical limit for an ideal gas of (quantum) molecules, and found q N /N ! for the classical or Boltzmann limit of the partition function of N indistinguishable molecules (see (17.74) on page 470). We also found the ﬁrst-order correction for either Fermi–Dirac or Bose–Einstein particles in terms of a virial coeﬃcient proportional to h3 (Eq. (17.68) on page 469). But these equations are only valid in the ideal gas case when the interaction between the particles can be ignored. In this section we shall consider a system of interacting particles and try to expand the partition function in powers of . We expect the zero-order term to be the classical limit, and we expect at least a third-order term to distinguish between FD and BE statistics. The approach to the classical limit of the quantum partition function

17.6 The classical approximation

2.5

473

Q

2 1.5 1 0.5 0

0.5

1

1.5

2

A,U/hν

2

U 1 0 A

−1 −2 0

0.5

1

0.5

1

1.5

2

S/k

2 1 0 −1 −2 0

1.5 2 Temperature kT/hν

Figure 17.4 Partition function Q, Helmholtz free energy A, energy U and entropy S for the harmonic oscillator as a function of temperature, for the quantum oscillator (drawn lines) and the classical oscillator (dashed lines). Temperature is expressed as kB T /hν, energies are in units of hν and entropy is in units of kB (ν being the oscillator frequency).

474

Review of statistical mechanics

was eﬀectively solved in the early 1930s. The original expansion in powers of was done by Wigner (1932), but without considering the symmetry properties that distinguish FD and BE statistics. The latter was solved separately by Uhlenbeck and Gropper (1932). Kirkwood (1933) gave a lucid combined derivation that found its way into most textbooks on quantum statistics. We shall not copy the derivation, but only give Kirkwood’s line of reasoning and the main result. We start with the expression of the quantum partition function (17.47) for a system of N identical particles:17 Q = tr exp(−βH),

(17.85)

where H is the Hamiltonian matrix in an arbitrary orthonormal complete set of basis functions. The basis functions must have the symmetry imposed by the particle characters, such as speciﬁed in (17.50). One convenient choice of basis function is the product of the wave functions for a single particle in a cubic periodic box with edge a and volume V , see (17.61). For this choice of plane wave functions 1 ˆ r) ∗ −β H( e ψk dr, (17.86) Q= ψk N! k with 2π nj , nj ∈ Z3 , a r = {r 1 , r 2 , . . . , r N }, r i ∈ box, 1 (∓1)P eiP[ j kj ·r j ] . ψk = √ V −N/2 N! P k = {k1 , k2 , . . . , kN }, kj =

(17.87) (17.88) (17.89)

The permutation P permutes indexes of identical particles, such as exchanging r i and r j , but it does not touch the indexing of k. The sign (∓1)P is negative when an odd number of fermion pairs are interchanged, and positive otherwise. The factor 1/N ! in the partition function needs some discussion. The product of plane waves does not form an orthogonal set, as each permutation within the set {k} produces an identical (except for the sign) wave function. Therefore sequences obtained by permutation should be omitted from the set {k}, which can be done, for example, by only allowing sequences for which the component k’s are ordered in non-decreasing order. If we allow all sequences, as we do, we overcount the sum by N ! and we should therefore divide the sum by N !.18 Note that this N ! has nothing to do with 17 18

For a mixture of diﬀerent types of particles, the equations are trivially modiﬁed. This is a rather subtle and usually disregarded consideration. The reader may check the

17.6 The classical approximation

475

√ the 1/ N ! in the deﬁnition of ψ(k), which is meant to normalize the wave function consisting of a sum of N ! terms. Since the box is large, the distance Δk = 2π/a between successive levels of each component of k is small and the sum over k can be replaced by an integral. When we also write p for k, we can replace the sum as: VN VN → dk = dp, (17.90) (2π)3N h3N k and obtain 1 1 ˆ P +P dp dr φ∗0 (P) e−β H(r ) φ0 (P), (∓1) (17.91) Q = 3N N !h N ! φ0 (P)

def

=

(i/)P[ j

e

P ,P

pj ·r j ] .

(17.92)

The problem is to evaluate the function u(r; P) = e−β H(r ) φ0 (P), def

ˆ

(17.93)

which is in general impossible because the two constituents of the hamilˆ = −(2 /2m) ∇2 and V (r) do not commute. If they would tonian K j j commute, we could write (see Section 14.6 on page 385) e−β H = e−βV e−β K , ˆ

ˆ

(17.94)

and evaluate u as u(r; P) = e−βV (r ) e−β

j(

p2j /2m) φ (P). 0

(17.95)

In (17.91) only the N ! terms with P = P survive and we would obtain the classical limit 1 Qcl = (17.96) dp dr e−βH(p,q ) . N !h3N In general, we need to solve u(r; P) (see (17.93)). By diﬀerentiating with respect to β, it s found that u satisﬁes the equation ∂u ˆ = −Hu, ∂β

(17.97)

which is the Schr¨ odinger equation in imaginary time: it/ being replaced by β. correctness for the case of two one-dimensional particles with plane waves k1 , k2 and observe that not only k1 k2 |k1 k2 = 1, but also k1 k2 |k2 k1 = ∓1. In fact, Kirkwood’s original article (1933) omitted this N !, which led to an incorrect omission of 1/N ! in the ﬁnal equations. Therefore his equations were still troubled by the Gibbs paradox. In a correction published in 1934 he corrected this error, but did not indicate what was wrong in the original article. The term is properly included by Huang (1987).

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Review of statistical mechanics

Kirkwood proceeded by writing u as u = w φ0 (P)e−βH(p,r ) ,

(17.98)

formulating a diﬀerential equation for w and then expanding w in a power series in . This yields (17.99) w = 1 + w1 + 2 w2 + O(3 ), ⎡ ⎤ iβ 2 ⎣ P pj · ∇j V ⎦ , w1 = − (17.100) 2m j ⎡ ⎤ 2 3 β β ⎣ 1 ∇2j V − (∇j V )2 + (P pj · ∇j )2 V ⎦ w2 = − 4m 6m m j

+

β4 8m2

(P

j

pj · ∇j V )2 .

j

(17.101)

j

Inserting this in (17.91) and integrating all correction terms over dp, ﬁrst the terms in the sum over permutations for which P = P (N ! terms) can be separated. All odd-order terms, which are antisymmetric in p, disappear by integration over p. Then the 12 N ! terms for which P and P diﬀer by the exchange of one pair of particles can be integrated. This is the leading exchange term; higher-order exchange terms will be neglected. The end result is 1 2πmkB T 3N/2 Q = dre−βV (r ) (1 + fcor ), N! h2

2 β 2 1 β 2 2 fcor = − ∇j V − (∇j V ) + O(4 ) 12 mj 2 j

−m r2 /β2 β j jk 1 + r jk · (∇j V − ∇k V ) + . . . .(17.102) e ∓ 2 j =k

We note that the 2 correction term was earlier derived by Wigner (1932); the exchange term had been derived by Uhlenbeck and Gropper (1932) in the slightly diﬀerent form an ensemble average of the product over particle 2 /2 ), and without the ﬁrst-order term in β pairs (i, j) of (1 ∓ exp(−mkB T rij in (17.102). What does this result mean in practice? First we see that the classical canonical partition function is given by 1 2πmkB T 3N/2 cl (17.103) Q = dre−βV (p,r ) N! h2

17.6 The classical approximation

=

1 N !h3N

dp

dre−βH(p,r ) .

477

(17.104)

This is the starting point for the application, in the following section, of statistical mechanics to systems of particles that follow classical equations of motion. We observe that this equation is consistent with an equilibrium probability density proportional to exp −βH(p, r) in an isotropic phase space p, r, divided into elementary units of area h with equal a priori statistical weights. There is one single-particle quantum state per unit of 6D volume h3 . The N ! means the following: If two identical particles (1) and (2) exchange places in phase space, the two occupations p(1)p (2)r(1)r (2) and p(2)p (1)r(2)r (1) should statistically be counted as one. The quantum corrections to the classical partition function can be expressed in several ways. The eﬀect of quantum corrections on thermodynamical quantities is best evaluated through the quantum corrections to the Helmholtz free energy A. Another view is obtained by expressing quantum corrections as corrections to the classical Hamiltonian. These can then be used to generate modiﬁed equations of motion, although one should realize that in this way we do not generate true quantum corrections to classical dynamics, but only generate some kind of modiﬁed dynamics that happens to produce proper quantum corrections to equilibrium phase-space distributions. First look at the quantum correction to the free energy A = −kB T ln Q. Noting that Q = Qcl (1 + fcor ), where · · · denotes a canonical ensemble average dp dr fcor exp(−βH) , fcor = dp dr exp(−βH)

(17.105)

(17.106)

we see, using ln(1 + x) ≈ x), that A = Acl − kB T fcor .

(17.107)

By partial integration the second derivative of V can be rewritten as: ∇2j V = β(∇j V )2 .

(17.108)

The 2 correction now reads A = Acl +

1 2 (∇j V )2 . 24(kB T )2 mj

(17.109)

j

The averaged quantity is the sum of the squared forces on the particles.

478

Review of statistical mechanics Helmholtz free energy /hν 1 0.75 0.5 0.25

Acl+corr

Aqu Acl

0 −0.25 −0.5

0.25

0.5

0.75

1 1.25 1.5 Temperature kT/hν

Figure 17.5 The classical Helmholtz free energy of the harmonic oscillator (longdashed), the 2 -correction added to the classical free energy (short-dashed) and the exact quantum free energy (solid).

The use of this 2 -Wigner correction is described in Section 3.5 on page 70. Quantum corrections for other thermodynamic functions follow from A. As an example, we give the classical, quantum-corrected classical, and exact quantum free energy for the harmonic oscillator in Fig. 17.5. The improvement is substantial. It is possible to include the 2 -term into the Hamiltonian as an extra potential term: V cor = −kB T f cor .

(17.110)

If done in this fashion, calculation of the force on particle i then requires a double summation over particles j and k, i.e., a three-body interaction.19 The inclusion in a dynamics simulation would be cumbersome and time consuming, while not even being dynamically correct. However, there are several ways to devise eﬀective interparticle interactions that will lead to the correct 2 -correction to the free energy when applied to equilibrium simulations. An intuitively very appealing approach is to consider each particle as a Gaussian distribution. The width of such a distribution can be derived from Feynman’s path integrals (see Section 3.3 on page 44) and leads to the Feynman–Hibbs potential, treated in Section 3.5 on page 70. 19

See, e.g., Huang (1987)

17.7 Pressure and virial

479

Next we consider the exchange term, i.e., the last line of (17.102). We drop the ﬁrst-order term in β, which is zero for an ideal gas and for high temperatures; it may however reach values of the order 1 for condensed phases. It is now most convenient to express the eﬀect in an extra correction potential: −mk T r2 /2 2 2 B ij ln 1 ∓ e−mkB T rij / ≈ ±kB T e . V cor = −kB T i<j

i<j

(17.111) The ﬁrst form comes from the equations derived by Uhlenbeck and Gropper (1932) (see also Huang, 1987); the second form is an approximation that is invalid for very short distances. This is an interesting result, as it indicates that fermions eﬀectively repel each other at short distance, while bosons attract each other. This leads to a higher pressure for fermion gases and a lower pressure for boson gases, as was already derived earlier (Eq. (17.68) on page 469). The interparticle correction potential can be written in terms of the de Broglie wave length Λ (see (17.69) on page 469): r 2 r 2

ij ij cor ≈ ±kB T exp −2π . ln 1 ∓ exp −2π Vij = −kB T Λ Λ i<j

(17.112) This exchange potential is not a true, but an eﬀective potential with the eﬀect of correcting equilibrium ensembles to ﬁrst-order for exchange eﬀects. The eﬀective potential “acts” only between identical particles. Figure 17.6 shows the form and size of the exchange potential. When compared to the interaction potential for a light atom (helium-4), it is clear that the eﬀects of exchange are completely negligible for temperatures above 15 K. In “normal” molecular systems at “normal” temperatures exchange plays no role at all.

17.7 Pressure and virial There are two deﬁnitions of pressure:20 one stems from (continuum) mechanics and equals the normal component of the force exerted on a surface per unit area; the other stems from the ﬁrst law of thermodynamics (16.24) and equals minus the partial derivative of the internal energy with respect to the volume at constant entropy, i.e., without exchange of heat. These deﬁnitions are equivalent for a system with homogeneous isotropic pressure: if a surface with area S encloses a volume V and a force p dS acts on every 20

The author is indebted to Dr Peter Ahlstr¨ om and Dr Henk Bekker for many discussions on pressure in the course of preparing a review that remained unpublished. Some of the text on continuum mechanics originates from Henk Bekker.

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Review of statistical mechanics

potential energy /kBT 3 5

10

15

20

25 30 K

2

1

fermions

0

−1

−2

bosons

0.25

0.5

0.75 1 1.25 1.5 interparticle distance r/Λ

Figure 17.6 Eﬀective exchange potential between fermions or bosons. The solid black curves are the ﬁrst form of (17.112); the dashed black curves are the approximate second form. The distance is expressed in units of the de Broglie wavelength √ Λ = h/ 2πmkB T . For comparison, the Lennard–Jones interaction for 4 He atoms, also expressed as a function of r/Λ, is drawn in gray for temperatures of 5, 10, 15, 20, 25 and 30 K.

surface element dS, moving the surface an inﬁnitesimal distance δ inwards, then an amount of work p Sδ = −p dV is done on the system, increasing its internal energy. But these deﬁnitions are not equivalent in the sense that the mechanical pressure can be deﬁned locally and can have a tensorial character, while the thermodynamic pressure is a global equilibrium quantity. In statistical mechanics we try to average a detailed mechanical quantity (based on an atomic description) over an ensemble to obtain a thermodynamic quantity. The question to be asked is whether and how a mechanical pressure can be locally deﬁned on an atomic basis. After that we can average over ensembles. So let us ﬁrst look at the mechanical deﬁnition in more detail.

17.7 Pressure and virial

481

"" "

" "

negative side "

positive side - normal dS = n dS

dF

"" "

" "

"

Figure 17.7 A force dF = σ · dS acts on the negative side of a surface element.

17.7.1 The mechanical pressure and its localization In a continuous medium a quantity related to to the local pressure, called stress, is given by a second-rank tensor σ(r), deﬁned through the following relation (see Fig. 17.7): the force exerted by the material lying on the positive side of a static surface element dS with normal n (which points from negative to positive side), on the material lying on the negative side of dS, is given by dF = σ(r) · n dS = σ(r) · dS, dFα = σαβ dSβ .

(17.113) (17.114)

β

The stress tensor is often decomposed into a diagonal tensor, the normal stress, and the shear stress τ which contains the oﬀ-diagonal elements. The force acting on a body transfers momentum into that body, according to Newton’s law. However, the stress tensor should be distinguished from the momentum ﬂux tensor Π, because the actual transport of particles also contributes to the momentum ﬂux. Also the sign diﬀers because the momentum ﬂux is deﬁned positive in the direction of the surface element (from inside to outside).21 The momentum ﬂux tensor is deﬁned as def

Παβ = −σαβ + ρvα vβ ,

(17.115)

Π = −σ + vJ ,

(17.116)

or

21

There is no sign consistency in the literature. We follow the convention of Landau and Lifschitz (1987).

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where ρ is the mass density and vJ is the dyadic vector product of the velocity v and the mass ﬂux J = ρv through the surface element. It is this momentum ﬂux tensor that can be identiﬁed with the pressure tensor, which is a generalization of the pressure. If Π is isotropic, Παβ = p δαβ , the force on a surface element of an impenetrable wall, acting from inside to outside, is normal to the surface and equals p dS. Is the pressure tensor, as deﬁned in (17.116) unique? No, it is not. The stress tensor itself is not a physical observable, but is observable only through the action of the force resulting from a stress. From (17.113) we see that the force F V acting on a closed volume V , as exerted by the surrounding material, is given by the integral over the surface S of the volume σ · dS. (17.117) FV = S

In diﬀerential form this means that the force density f (r), i.e., the force acting per unit volume, is given by the divergence of the stress tensor: f (r) = ∇ · σ(r).

(17.118)

Thus only the divergence of the stress tensor leads to observables, and we are free to add any divergence-free tensor ﬁeld σ 0 (r) to the stress tensor without changing the physics. The same is true for the pressure tensor Π. Without further proof we note that, although the local pressure tensor is not unique, its integral over a ﬁnite conﬁned system, is unique. The same is true for a periodic system by cancelation of surface terms. Therefore the average pressure over a conﬁned or periodic system is indeed unique. Turning from continuum mechanics to systems of interacting particles, we ask the question how the pressure tensor can be computed from particle positions and forces. The particle ﬂux component vJ of the pressure tensor is straightforward because we can count the number of particles passing over a deﬁned surface area and know their velocities. For the stress tensor part all we have is a set of internal forces F i , acting on particles at positions r i .22 From that we wish to construct a stress tensor such that ∇ · σ(r) = F i δ(r − r i ). (17.119) i

Of course this construction cannot be unique. Let us ﬁrst remark that a solution where σ is localized on the interacting particles is not possible for the simple reason that σ cannot vanish over a closed surface containing a 22

Here we restrict the pressure as resulting from internal forces, arising from interactions within the system. If there are external forces, e.g., external electric or gravitational ﬁelds, such forces are separately added.

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483

particle on which a force is acting, because the divergence inside the closed surface is not zero. As shown by Schoﬁeld and Henderson (1982), however, it is possible to localize the stress tensor on arbitrary line contours C0i running from a reference point r 0 to the particle positions r i : F i,α δ(r − r c )(dr c )β . (17.120) σαβ (r) = − C0i

i

For each particle this function is localized on, and equally distributed over, the contour C0i . Taking the divergence of σ we can show that (17.119) is recovered: F i∇ · δ(r − r c ) dr c ∇ · σ(r) = − =

i

C0i

F i [δ(r i ) − δ(r 0 )]

i

=

F i δ(r i ).

(17.121)

i

Proof The ﬁrst step follows from the three-dimensional generalization of b b d d f (ξ − x) dx = −f (ξ − b) + f (ξ − a), (17.122) f (ξ − x) dx = − dξ a a dx the last step uses the fact that i F i = 0 for internal forces. If we integrate the stress tensor over the whole (conﬁned) volume of the system of particles, only the end points in the line integral survive and we obtain the sum of the dyadic products of forces and positions: σ d3 r = − F i (r i − r 0 ) = − F i ri , (17.123) V

i

i

which is independent of the choice of reference position r 0 . The introduction of a reference point is undesirable as it may localize the stress tensor far away from the interacting particles. When the forces are pair-additive, the stress tensor is the sum over pairs; for each pair i, j the two contours from the reference position can be replaced by one contour between the particles, and the reference position cancels out.

-

i t

Cij t j

Q k Q

C0iQ

t

0

C

0j

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(a)

(c)

(b)

(d)

Figure 17.8 Four diﬀerent contours to localize the stress tensor due to interaction between two particles. Contour (b) is the minimal path with optimal localization.

The result is σαβ =

i<j

δ(r − r c ) (dr c )β ,

Fijα

(17.124)

Cij

with integral σ αβ d3 r = Fiα (rjβ − riβ ) = −Fiα riβ − Fjα rjβ ,

(17.125)

V

consistent with (17.123). Here Fijα is the α component of the force acting on i due to the interaction with j. The path between the interacting particle pair is irrelevant for the pressure: Fig. 17.8 gives a few examples including distribution over a collection of paths (a), the minimal path, i.e., the straight line connecting the particles (b), and curved paths (c, d) that do not conﬁne the localization to the minimal path. Irving and Kirkwood (1950) chose the straight line connecting the two particles as a contour, and we recommend that choice.23

17.7.2 The statistical mechanical pressure Accepting the mechanical deﬁnition of the pressure tensor as the momentum ﬂux of (17.116), we ask what the average pressure over an ensemble is. First we average over the system, and then over the ensemble of systems. The average over the whole system (we assume our system is conﬁned in space) is given by the integral over the volume, divided by the volume (using dyadic vector products): 1 1 ΠV = − σ(r)d3 r + mi v i v i , (17.126) V V V i

23

This choice is logical but not unique, although Wajnryb et al. (1995) argue that additional conditions make this choice unique. The choice has been challenged by Lovett and Baus (1997; see also Marechal et al., 1997) on the basis of a local thermodynamic pressure deﬁnition, but further discussion on a physically irrelevant choice seems pointless (Rowlinson, 1993). For another local deﬁnition see Zimmerman et al. (2004).

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485

or, with (17.123):

ΠV V =

F i ri +

i

mi v i v i .

(17.127)

i

In a dynamic system this instantaneous volume-averaged pressure is a ﬂuctuating function of time. We remark that (in contrast to the local tensor) this averaged tensor is symmetric, because in the ﬁrst term the diﬀerence between an oﬀ-diagonal element and its transpose is a component of the total torque on the system, which is always zero in the absence of external forces. The second term is symmetric by deﬁnition. Finally, the thermodynamic pressure tensor P which we deﬁne as the ensemble average of the volume-averaged pressure tensor, is given by PV = F i r i + mi v i v i , (17.128) i

i

where the angular brackets are equilibrium ensemble averages, or because of the ergodic theorem, time averages over a dynamic equilibrium system. P is a “sharp” symmetric tensor. For an isotropic system the pressure tensor is diagonal and its diagonal elements are the isotropic pressure p: pV =

1 1 2 tr P V = F i · r i + Ekin . 3 3 3

(17.129)

i

The ﬁrst term on the right-hand side relates to the virial of the force, already deﬁned by Clausius (see, e.g., Hirschfelder et al., 1954) and valid for a conﬁned (i.e., bounded) system: def

Ξ = −

1 F i · r i . 2

(17.130)

i

Here F i is the total force on particle i, including external forces. The resulting virial Ξ is the total virial, which can be decomposed into an internal virial due to the internal forces and an external virial, for example caused by external forces acting on the boundary of the system in order to maintain the pressure. Written with the Clausius virial, (17.129) becomes 2 pV = (Ekin − Ξint ). 3

(17.131)

This relation follows also directly from the classical virial theorem: Ξtot = Ekin

(17.132)

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which is valid for a bounded system.24 This virial theorem follows also from the generalized equipartition theorem, treated in Section 17.10 on page 503. Since the external virial due to an external force −p dS acting on any surface element dS equals 1 Ξext = p 2

3 r · dS = pV, 2 S

(17.133)

the virial theorem immediately yields (17.131) (see Exercise 17.6).

Periodic boundary conditions The virial expression and the pressure equations given above are valid for a bounded system, but not for an inﬁnite system with periodic boundary conditions, as so often used in simulations. Under periodic boundary conditions the force on every particle may be zero while the system has a non-zero pressure: imagine the simple case of one particle in a cubic box, interacting symmetrically with six images. Then i F i r i is obviously zero. For forces that can be written as a sum of pair interactions this problem is easily remedied by replacing i F i r i by a sum over pairs:25 i

F i ri =

F ij r ij ,

(17.134)

i<j

where F ij is the force on i due to j and r ij = r i −r j . This sum can be taken over all minimum-image pairs if no more than minimum images are involved in the interaction. For interactions extending beyond minimum images, the pressure tensor can be evaluated over the volume of a unit cell, according to Irving and Kirkwood’s distribution over a straight line, using the fraction of the line lying within the unit cell. Consider the simple case, mentioned above, with a single particle in a cubic unit cell of size a × a × a.

24 25

See, e.g., Hirschfelder et al. (1954). A very extensive review on the virial theorem, including a discussion on the quantum-mechanical form, has been published by Marc and McMillan (1985). Erpenbeck and Wood (1977).

17.7 Pressure and virial

Assume that each of its six images exert a force F on the particle, with a zero vector sum. Each contribu tion F a to the sum i F i · r i counts for 0.5 since the interaction line lies for 50% in the unit cell, so the total sum is 3F a, and the virial contribution to the pressure (17.129) equals F a/V = F/a2 . This is correct, as we can see that this pressure, if acting externally on one side with area a2 , just equals the force acting “through” that plane.

487

F 6 u 1

p a2

?????

F3 u

F4

u-F4 6 F2 ?

u-

F2

F1

u ?F3

Now consider the interaction between two particles i and j (Fig. 17.9). In a lattice sum interactions between all image pairs of both i and j are included; note that there are always sets of equivalent pairs, e.g., in the case of a single shift with n = (010) (Fig. 17.9b): r i − (rj + Tn) = (r i − Tn) − rj ,

(17.135)

where T is the transformation matrix from relative coordinates in the unit cell to cartesian coordinates (see (6.3) on page 143), i.e., a matrix of which the columns are the cartesian base vectors of the unit cell, and n ∈ Z3 . Figure 17.9 shows three examples of image pairs, with one, two and three equivalent pairs, respectively. If we add up the fractions of the interaction lines that run through the unit cell, we obtain just one full interaction line, so the contribution of that set of pairs to the sum F ij · r ij is given by F ijn · (r i − rj − Tn). Note that each set of equivalent pairs contributes only once to the total energy, to the force on i, to the force on j and to the virial contribution to the pressure. Replacing the dot product by a dyadic product, the scalar contribution is generalized to a tensorial contribution. Summarizing, for a lattice sum of isotropic pair interactions vij (r), the total potential energy, the force on particle i and the instantaneous pressure tensor (see (17.127)) are given by Epot =

1 vij (rijn ), 2

(17.136)

i,j,n

Fi =

j,n

F ijn , F ijn =

dvij rijn , dr |rijn |

(17.137)

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(a) (b)

(c)

Figure 17.9 Three examples of “equivalent image pairs” of two interacting particles (open circle and open square in the shaded unit cell; images are ﬁlled symbols). For each image pair the fraction of the dashed interaction line lying in the unit cell, which contributes to the pressure, is drawn as a solid line.

ΠV =

1 F ijn r ijn + mi v i v i . 2

(17.138)

rijn = |r ijn |,

(17.139)

r ijn = r i − r j − Tn.

(17.140)

i,j,n

i

Here we use the notation

Note that the volume V is equal to det T. The sum is taken over all particles i and all particles j in the unit cell, and over all sets of three integers n : n ∈ Z3 . This includes i and its images; the prime in the sum means that j = i is excluded when n = 0. The factor 12 prevents double counting of pairs, but of course summation over i, j; i = j can be restricted to all i with all j > i because i and j can be interchanged while replacing n by −n. The factor 12 must be maintained in the iin summation. The summation over images may well be conditionally convergent, as is the case for Coulomb interactions. This requires a speciﬁed summation order or

17.7 Pressure and virial

489

special long-range summation techniques, as discussed in Section 13.10 on page 362. The equation for the pressure (17.138) is usually derived for the canonical ensemble from the equation p = −kB T (∂ ln Q/∂V )T or a tensorial variant that implies diﬀerentiating to the components of the transformation tensor T. The volume dependence in the partition function is then handled by transforming to scaled coordinates ρ (r = Tρ) which concentrates the volume dependence on T. The volume change is “coupled” to all particles in the system, rather then to particles on the surface as a volume change in a real experiment would do. This, of course, is also a choice that inﬂuences the instantaneous pressure, but not the ensemble-averaged pressure. Equation (17.138) is obtained.26 It is interesting that this coupling to all particles is equivalent to Irving and Kirkwood’s choice for the local pressure deﬁnition. Explicit equations for use with Ewald summation have been given by Nos´e and Klein (1983) and for use with the Particle Mesh Ewald method by Essmann et al. (1995). Pressure from center-of-mass attributes Thus far we have considered detailed atomic motion and forces on atoms to determine the pressure. However, pressure is a result of translational motion and forces causing translational motion. For a system consisting of molecules it is therefore possible to consider only the center-of-mass (c.o.m.) velocities and the forces acting on the c.o.m. Equation (17.128) is equally valid when F i are the forces acting on the c.o.m. and r i and v i denote c.o.m coordinates and velocities. Somehow the extra “intramolecular” virial should just cancel the intramolecular kinetic energy. Can we see why that is so?27 Consider a system of molecules with c.o.m. position Ri , each consisting of atoms with positions r ik (see Fig. 17.10). A total force F ik is acting on this atom. Denoting the mass of the molecule as Mi = k mik , the c.o.m. coordinate is given by Mi R i = mik r ik . (17.141) k

Now we deﬁne intramolecular coordinates sik of each atom with respect to 26

27

Pressure calculations on the basis of Hamiltonian derivatives and their use in constant pressure algorithms have been pioneered by Andersen (1980) for isotropic pressure and by Parrinello and Rahman (1980, 1981) for the pressure tensor. A good summary is to be found in Nos´e and Klein (1983) and an extensive treatment of pressure in systems with constraints has been given by Ciccotti and Ryckaert (1986). We roughly follow the arguments given in an appendix by Ciccotti and Ryckaert (1986).

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atom k mass mki ski rki Rj molecule i mass Mi

molecule j mass Mj

Ri origin

Figure 17.10 Deﬁnition of atoms clustered in molecules for discussion of the c.o.m. virial.

the c.o.m.: def

sik = r ik − Ri , with the obvious relation

mik sik = 0.

(17.142)

(17.143)

k

The total virial (see (17.130)) on an atomic basis can be split into a c.o.m. and an intramolecular part: 1 i i 1 i F k r k = − F k (Ri + sik ) Ξtot = − 2 2 i i k k 1 1 intra = − F i Ri − F ik sik = Ξcom tot + Ξtot . (17.144) 2 2 i

k

The forces are the total forces acting on the atom. Likewise we can split the kinetic energy: 1 i i i 1 i ˙ ˙ i + s˙ i ) mk r˙ k r˙ k = mk (Ri + s˙ ik )(R Ekin = k 2 2 i i k k 1 1 ˙ iR ˙i+ = Mi R mik s˙ ik s˙ ik 2 2 =

i com Ekin

i

+

intra Ekin .

k

(17.145)

intra If we can prove that Ξintra tot = Ekin , then we have proved that the pressure computed from c.o.m. forces and velocities equals the atom-based pressure.

17.7 Pressure and virial

491

The proof is simple and rests on the fact that neither the internal coordinates nor the internal velocities can grow to inﬁnity with increasing time. First realize that F ik = mik r¨ ik ; then it follows that for every molecule (we drop the superscripts i and use (17.143)) ¨ k sk . F k sk = − mk s (17.146) − k

k

Ensemble-averaging can be replaced by time averaging: 1 T ¨k sk dt F k sk = − lim mk s − T →∞ T 0 k k 1 = mk s˙ k s˙ k − lim [sk s˙ k (T ) − sk s˙ k (0)].(17.147) T →∞ T k

Since the last term is a ﬂuctuating but bounded quantity divided by T , the limit T → ∞ is zero and we are left with the equality of intramolecular virial and kinetic energy, if averaged over an equilibrium ensemble. The reasoning is equivalent to the proof of the virial theorem (17.132) (Hirschfelder et al., 1954). Note that the “molecule” can be any cluster of particles that does not fall apart in the course of time; there is no requirement that the cluster should be a rigid body. The subtlety of virial-kinetic energy compensation is nicely illustrated by the simple example of an ideal gas of classical homonuclear diatomic molecules (bond length d) with an internal quadratic potential with force constant k. We can calculate the pressure from the total kinetic energy minus the internal atomic virial, but also from the kinetic energy of the c.o.m. minus the molecular virial. So the virial of the internal forces, which can only be due to a force acting in the bond direction, must be compensated by the intramolecular kinetic energy. Of the total of six degrees of freedom three are internal: the bond vibration and two rotational degrees of freedom, together good for an average of 32 kB T kinetic energy. The harmonic bond vibrational degree of freedom has an average kinetic energy of 12 kB T , which is equal to the average potential energy 12 k(d − d0 )2 . Therefore the contribution to the virial is 12 F d = 12 kd(d − d0 ) = 12 k(d − d0 )2 = 12 kB T , which exactly cancels the average kinetic energy of the bond vibration. But how about the contribution of the two rotational degrees of freedom? They have an average kinetic energy of kB T , but where is the virial compensation? The answer is that rotation involves centrifugal forces on the bond, which is then stretched, causing a compensating elastic force in the bond direction. That force causes a contribution to the intramolecular virial that exactly compensates the rotational kinetic energy (see exercise 17.8).

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With that problem solved, the next question is what happens if the bond length is treated as a constraint? In that case there is no kinetic energy in the bond vibration and their is no contribution to the virial due to vibrational motion. But there still is rotational kinetic energy and that is still compensated by the contribution to the virial of the constraint force. So when constraints are used in an atomic description, the constraint forces must be computed and accounted for in the atomic virial. Constraint forces are the forces that compensate components of other forces in the constraint directions; they follow directly from the constraint computation (see Section 15.8 on page 417).

17.8 Liouville equations in phase space A classical system of particles that evolves under Hamiltonian equations of motion (see Section 15.3 on page 399) follows a trajectory in the 2ndimensional space of all its (generalized) coordinates qi , i = 1, . . . n and conjugate momenta pi i = 1, . . . n, called phase space. Here n is the number of degrees of freedom, which equals 3× the number of particles minus the number of constrained degrees of freedom, if any. A particular conﬁguration of all coordinates and momenta at time t deﬁnes a point in phase space. Often it is convenient to denote the 2n-dimensional vector (q1 , . . . qn , p1 , . . . pn )T by the vector z (z ∈ R2n ).28 We consider for the time being systems that obey a ﬁrst-order diﬀerential equation, so that the evolution of a point in phase space is a deterministic initial value problem. Consider evolution under the Hamilton equations (15.9): ∂H(q, p) , ∂pi ∂H(q, p) . p˙i = − ∂qi q˙i =

In symplectic notation, and introducing the matrix 0 1 , L0 = −1 0

(17.148) (17.149)

(17.150)

where 0 is a n × n all-zero matrix, and 1 is a n × n diagonal unit matrix, 28

This notation is referred to as the symplectic notation. A symplectic mapping is an area (and volume) conserving mapping, such as the transformation of z for systems obeying Hamilton’s equations. A symplectic algorithm is an algorithm to solve the time evolution that conserves area and volume in phase space.

17.8 Liouville equations in phase space

493

Hamilton’s equations can be written as z˙ = L0

∂H . ∂z

(17.151)

Another notation is writing the operation on z on the right-hand side of (17.151) as an operator : ˆ z˙ = iLz.

(17.152)

Lˆ is the Liouville operator, which we will deﬁne below in the context of the rate of change of density in phase space. The rate equation (17.152) is more general than its Hamiltonian form of (17.151), and may for example contain the eﬀects of external time-dependent ﬁelds. The operator is then called the generalized Liouville operator. The rate equation can be formally integrated to yield a time evolution equation for the point z in phase space: t ˆ ) dτ ] z(0), z(t) = exp[ iL(τ (17.153) 0

which for time-independent operators reduces to z(t) = eiLt z(0). ˆ

(17.154)

These equations are formally elegant but do not help solving the equations of motion. Exponential operators and the integral over a time-dependent operator must all be carefully deﬁned to be meaningful (see Section 14.6 on exponential operators on page 385). In statistical mechanics we deal not with single trajectories, but with distribution functions f (z, t) in phase space. We are interested in the time evolution of such distribution functions and in the equilibrium distributions corresponding to the various ensembles. When we know the distribution function, we can determine the observable ensemble average of any variable that is known as a function of the point in phase space: A(t) = A(z) f (z, t) dz. (17.155) In order to ﬁnd a diﬀerential equation for the time evolution of f (z, t) we try to ﬁnd the time derivative ∂f /∂t. Since f concerns a probability distribution, its integrated density must be conserved, and any change in density ˙ The conservation law in 2n must result from in- or outgoing ﬂux J = f z. dimensions is similar to the continuity equation in ﬂuid dynamics (see (9.3)

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on page 282): ∂f (z, t) ˙ = −z˙ · ∇f − f ∇ · z, ˙ = −∇ · J = −∇ · (f z) ∂t

(17.156)

where ∇ stands for the gradient operator in z: (∂/∂z1 , . . . , ∂/∂z2n ). Writing the time derivative as the material or Lagrangian derivative D/Dt, i.e., the derivative seen when traveling with the ﬂow (see page 282), we see that Df def ∂f ˙ + z˙ · ∇f = −f ∇ · z. = Dt ∂t

(17.157)

This equation is often referred to as the generalized Liouville equation. For a Hamiltonian system the term on the right-hand side is zero, as we see using (17.151):

2n ∂H ∂2H L0ij = ∇ · z˙ = ∇ · L0 ∂z ∂zi ∂zj i,j=1 n ∂2H ∂2H = − = 0. ∂qi ∂pj ∂pi ∂qj

(17.158)

i=1

This is the proper Liouville equation, which is very important in statistical mechanics. It states that for a Hamiltonian system the (probability) density in phase space does not change with time. This also means that a volume in phase space does not change with time: if one follows a bundle of trajectories that start in an initial region of phase space, then at a later time these trajectories will occupy a region of phase space with the same volume as the initial region.29 This is also expressed by saying that the Hamiltonian probability ﬂow in phase space is incompressible. If the volume does not change, neither will a volume element used for integration over phase space. We have to be careful what we call a volume element. Normally we write the volume element somewhat loosely by a product dz or d2n z or dz1 . . . dz2n or Π2n i=1 dzi , while we really mean the volume spanned by the local displacement vectors corresponding to the increments dzi . These displacement vectors are proportional, but not necessarily equal to zi . The volume spanned by the local displacement vectors only equals their product when these vectors are orthogonal; in general the volume is equal to the determinant of the matrix formed by the set of displacement vectors. The proper name for such a volume is the wedge product, but we 29

We skip the intricate discussion on the possibility to deﬁne such regions, which relates to the fact that diﬀerent trajectories can never cross each other. Even more intricate is the discussion on the possible chaotic behavior of Hamiltonian dynamical systems that destroys the notion of conserved volume.

17.8 Liouville equations in phase space

495

shall not use the corresponding wedge notation. The volume element is written as √ dV = gdz1 · · · dz2n , (17.159) where g is the determinant of the metric tensor gij , which deﬁnes the metric of the space: the square of the length of the displacement ds caused by dz1 , . . . , dz2n is determined by (ds)2 = i,j gij dzi dzj .30 . When coordinates are transformed from z(0) to z(t), the transformation is characterized by a transformation matrix J and the volume element transforms with the Jacobian J of the transformation, which is the determinant of J: g(t) dz1 (t) · · · dz2n (t) = g(0) dz1 (0) · · · dz2n (0), (17.160) dz1 (t) · · · dz2n (t) = J dz1 (0) · · · dz2n (0),

(17.161)

dz(t) = J dz(0),

(17.162)

J = det J, J = g(t)/g(o).

(17.163) (17.164)

The Liouville equation (17.158) implies that the Jacobian of the transformation from z(0) to z(t) equals 1 for Hamiltonian systems. Hence the volume element in phase space, which is denoted by the product dz = dz1 (t) · · · dz2n (t), is invariant under a Hamiltonian or canonical transformation. A canonical transformation is symplectic, meaning that the so-called two-form dq ∧ dp = i dqi ∧ dpi of any two vectors dq and dp, spanning a two-dimensional surface S, is invariant under the transformation.31 This property is important in order to retain the notion of probability density in phase space f (z) dz. In practice, time evolutions are not always Hamiltonian and the probability ﬂow could well loose its incompressibility. The question how the Jacobian (or the metric tensor) develops in such cases and inﬂuences the distribution functions has been addressed by Tuckerman et al. (1999). We’ll summarize their results. First deﬁne the phase space compressibility κ: def

κ(z) = ∇ · z˙ =

∂ z˙i i

30

31

∂zi

,

(17.165)

Consider polar coordinates (r, θ, φ) of a point in 3D space: changing θ to θ + dθ causes a displacement vector of length r dθ. Changing φ to φ + dφ causes a displacement vector of length r sin θ dφ. What is the metric tensor for polar coordinates and what is the square root of its determinant and hence the proper volume element? The two-form is the sum of areas of projections of the two-dimensional surface S onto the qi − pi planes. See, e.g., Arnold (1975).

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which is the essential factor on the right-hand side of the generalized Liouville equation (17.157). As we have seen, κ = 0 for incompressible Hamiltonian ﬂow. Tuckerman et al. then proceed to show that the Jacobian J of the transformation from z(0) to z(t) obeys the diﬀerential equation dJ = J κ(z), dt with solution

J(t) = exp

(17.166)

t

κ[z(τ )] dτ

.

(17.167)

0

If a function w(z) is deﬁned for which w˙ = κ, then J(t) = ew(z(t))−w(z(0) ,

(17.168)

and (17.169) e−w(z(t)) dz1 (t) · · · dz2n (t) = e−w(z(0)) dz1 (0) · · · dz2n (0). √ Hence this modiﬁed volume element, with g = e−w(z) , is invariant under the non-Hamiltonian equations of motion. This enables us to compute equilibrium distribution functions generated by the non-Hamiltonian dynamics. Examples are given in Section 6.5 on page 194. Let us return to Hamiltonian systems for which the Liouville equation (17.158) is valid. The time derivative of f , measured at a stationary point in phase space, is ∂f ˆ = −z˙ · ∇f = −iLf, (17.170) ∂t where the Liouville operator is deﬁned as32 iLˆ

2n ∂ ∂H ∂ = L0ij ∂zi ∂zj ∂zi i=1 i,j=1 n ∂H ∂ ∂H ∂ . = − ∂pj ∂qi ∂qj ∂pi

def

=

z˙ · ∇ =

2n

z˙i

(17.171)

i,j=1

This sum is called a Poisson bracket; if applied to a function f , it is written as {H, f }. We shall not use this notation. Assuming a Hamiltonian that does not explicitly depend on time, the formal solution is f (z, t) = e−itL f (z, 0). ˆ

32

(17.172)

The convention to write the operator as iLˆ and not simply Lˆ is that there is a corresponding operator in the quantum-mechanical evolution of the density matrix and the operator now is hermitian.

17.9 Canonical distribution functions

497

Note that the time-diﬀerential operator (17.170) for the space phase density has a sign opposite to that of the time-diﬀerential operator (17.152) for a point in phase space.

17.9 Canonical distribution functions In this section we shall consider the classical distribution functions for the most important ensemble, the canonical ensemble, for various cases. The cases concern the distributions in phase space and in conﬁguration space, both for cartesian and generalized coordinates and we shall consider what happens if internal constraints are applied. In general phase space the canonical (NVT) equilibrium ensemble of a Hamiltonian system of n degrees of freedom (= 3N for a system without constraints) corresponds to a density f (z) in phase space proportional to exp(−βH): f (z) =

exp[−βH(z)] . exp[−βH(z)] d2n z

(17.173)

The classical canonical partition function for N particles (for simplicity taken as identical; if not, the N ! must be modiﬁed to a product of factorials for each of the identical types) is 1 Q = 3N exp[−βH(z)] d2n z. (17.174) h N! The Hamiltonian is the sum of kinetic and potential energy.

17.9.1

Canonical distribution in cartesian coordinates

We now write r i , pi = mi r˙i (i = 1, . . . , N ) for the phase-space coordinates z. The Hamiltonian is given by H=

N p2i + V (r). 2mi

(17.175)

i=1

The distribution function (17.173) can now be separately integrated over momentum space, yielding a conﬁgurational canonical distribution function f (r) = V

exp[−βV (r)] exp[−βV (r)] dN r

(17.176)

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Review of statistical mechanics

while the classical canonical partition function is given by integration of (17.174) over momenta: 1 2πkB T 3N/2 N 3/2 Πi=1 mi e−βV (r ) dN r. (17.177) Q= N! h2 V Using the deﬁnition of the de Broglie wavelength Λi (17.69): Λi = √

h , 2πmi kB T

the partition function can also be written as 1 N −3 Πi=1 Λi e−βV (r ) dN r. Q= N! V

(17.178)

17.9.2 Canonical distribution in generalized coordinates In generalized coordinates (q, p) the kinetic energy is a function of the coordinates, even if the potential is conservative, i.e., a function of coordinates only. The Hamiltonian now reads (see 15.24) on page 402): H = pT M−1 (q)p + V (q),

(17.179)

where M is the mass tensor (see (15.16) on page 401) Mkl =

N i=1

mi

∂r i ∂r i · . ∂qk ∂ql

(17.180)

In cartesian coordinates the mass tensor is diagonal with the masses mi on the diagonal (each repeated three times) and the integration over momenta can be carried out separately (see above). In generalized coordinates the integration over momenta yields a q-dependent form that cannot be taken out of the integral: 1 2πkB T 3N/2 Q= |M|1/2 e−βV (q) dn q, (17.181) N! h2 V where we use the notation |A| for the determinant of A. So, expressed as integral over generalized conﬁgurational space, there is a weight factor (det M)1/2 in the integrand. The integration over momenta is obtained by transforming the momenta with an orthogonal transformation so as to obtain a diagonal inverse mass tensor; integration then yields the square root of the product of diagonal elements, which equals the square root of the determinant of the original inverse mass matrix. It is also possible to arrive at this equation by ﬁrst integrating over momenta in cartesian coordinates

17.9 Canonical distribution functions

499

and then transforming from cartesian (x1 , . . . xn ) to generalized (q1 , . . . , qn ) coordinates by a transformation J: Jik =

∂xi ∂qk

(17.182)

with Jacobian J(q) = |J|, yielding 1 2πkB T 3N/2 N 3/2 Πi=1 mi J(q) e−βV (q) dn q. Q= N! h2 V

(17.183)

Apparently, 3/2

|M|1/2 = ΠN i=1 mi J,

(17.184)

as follows immediately from the relation between mass tensor (17.180) and transformation matrix: M = JT Mcart J,

(17.185)

where Mcart is the diagonal matrix of masses, so that 3 |M| = |J|2 ΠN i=1 mi .

(17.186)

The result is that the canonical ensemble average of a variable A(q) is given by A(q)|M|1/2 exp[−βV (q)] dq . (17.187) A = |M|1/2 exp[−βV (q)] dq 17.9.3 Metric tensor eﬀects from constraints The question addressed here is what canonical equilibrium distributions are generated in systems with constraints, and consequently, how averages should be taken to derive observables. Such systems are usually simulated in cartesian coordinates with the addition of Lagrange multipliers that force the system to remain on the constraint hypersurface in phase space; the dynamics is equivalent to that in a reduced phase space of non-constrained generalized coordinates. The result will be that there is an extra weight factor in the distribution function. This result has been obtained several times in the literature, for example see Frenkel and Smit (1996) or Ciccotti and Ryckaert (1986). Consider generalized coordinates q1 , . . . , q3N = (q q ) that are chosen in such a way that the last nc coordinates q are to be constrained, leaving the ﬁrst n = 3N − nc coordinates q free. The system is then restricted to n degrees of freedom q . We ﬁrst consider the fully, i.e., mathematically, constrained case, where q = c, c being a set of constants, and without any

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kinetic energy in the constrained coordinates. Writing the mass tensor in four parts corresponding to q and q : F D , (17.188) M= DT C the kinetic energy now equals 1 K = q˙ T Fq˙ , 2

(17.189)

p = Fq˙ ,

(17.190)

leading to conjugate momenta

which diﬀer from the momenta in full space. The canonical average of a variable A(q ) obtained in the constrained system is A(q )|F|1/2 exp[−βV (q )] dq . (17.191) Ac = |F|1/2 exp[−βV (q )] dq The same average is obtained from constrained dynamics in cartesian coordinates: A(r)Πs δ(σs (r)) exp[−βV (r)] dr , (17.192) Ac = Πs δ(σs (r)) exp[−βV (r)] dr where σs (r) = 0, s = 1, . . . , nc , are the constraint equations that remain satisﬁed by the algorithm. Compare this with a classical physical system where q are near constraints that only negligibly deviate from constants, for example restrained by stiﬀ oscillators. The diﬀerence with mathematical constraints is that the near constraints do contribute to the kinetic energy and have an additional potential energy Vc (q ) as well. The latter is a harmonic-like potential with a sharp minimum. In order to obtain averages we need the full conﬁguration space, but within the integrand we can integrate over q : (17.193) Q = exp[−βVc (q )] dq . The average over the near-constraint canonical ensemble is A(q )|M (q , c)|1/2 Q exp[−βV (q ) dq ] . Anc = |M (q , c)|1/2 Q exp[−βV (q ) dq ]

(17.194)

Q may depend on q : for example, for harmonic oscillators Q depends on the force constants which may depend on q . But this dependence is weak and often negligible. In that case Q drops out of the equation and the weight factor is simply equal to |M|1/2 . Since |F | = |M |, the weight factor

17.9 Canonical distribution functions

501

in the constraint ensemble is not equal to the weight factor in the classical physical near-constraint case. Therefore the constraint ensemble should be corrected by an extra weight factor (|M|/|F|)1/2 . This extra weight factor can also be expressed as exp[−βvc (q )] with an extra potential 1 |M| . vc (q ) = − kB T ln 2 |F|

(17.195)

The ratio |M|/|F| seems diﬃcult to evaluate since both M and F are complicated large-dimensional matrices, even when there are only a few constraints. But a famous theorem by Fixman (1979) saves the day: |M| |Z| = |F|,

(17.196)

where Z is the (q q ) part of the matrix M−1 : 1 ∂q ∂q X Y s −1 , Zst (q ) = · t. M = T Y Z mi ∂r i ∂r i

(17.197)

i

Z is a low-dimensional matrix which is generally easy to compute. We ﬁnd for the extra weight factor |Z|−1/2 , or 1 vc (q ) = kB T ln |Z|.. (17.198) 2 The corrected constrained ensemble average of an observable A can be expressed in cartesian coordinates as |Z|−1/2 A(r)Πs δ(σs (r)) exp[−βV (r)] dr A = . (17.199) |Z|−1/2 Πs δ(σs (r)) exp[−βV (r)] dr For completeness we give the ingenious proof of Fixman’s theorem. Proof From −1

M

M=

X Y YT Z

F D DT C

= 1,

we see that XF + YDT = 1, YT F + ZDT = 0. Consider X F 0 F 0 −1 = MM =M T T YT D 1 D 1 XF + YDT Y =M = M YT F + ZDT Z

F 0 DT 1 1 Y . (17.200) 0 Z Y Z

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Review of statistical mechanics

Hence |F| = |M| |Z|. This eﬀect is often referred to as the metric tensor eﬀect, which is not a correct name, since it is not the metric tensor proper, but the mass(-metric) tensor that is involved. The eﬀect of the mass tensor is often zero or negligible. Let us consider a few examples: • A single distance constraint between two particles: q = r12 = |r 1 − r 2 |. The matrix Z has only one component Z11 : Z11 =

1 ∂r12 ∂r12 1 ∂r12 ∂r12 1 1 · + · = + . m1 ∂r 1 ∂r 1 m2 ∂r 2 ∂r 2 m1 m2

(17.201)

This is a constant, so there is no eﬀect on the distribution function. • A single generalized distance constraint that can be written as q = R = | i αi r i |, with αi being constants. For this case Z11 = i αi2 /mi is also a constant with no eﬀect. • Two distance constraints r12 and r32 for a triatomic molecule with an angle φ between r 12 and r 32 . The matrix Z now is a 2 × 2 matrix for which the determinant appears to be (see Exercise 17.9): 1 1 1 1 1 − 2 cos2 φ. + + (17.202) |Z| = m1 m1 m2 m3 m2 This is a nonzero case, but the weight factor is almost constant when the bond angle is nearly constant. More serious eﬀects, but still with potentials not much larger than kB T , can be expected for bond length and bond angle constraints in molecular chains with low-barrier dihedral angle functions. It seems not serious to neglect the mass tensor eﬀects in practice (as is usually done). It is, moreover, likely that the correction is not valid for the common case that constrained degrees of freedom correspond to high-frequency quantum oscillators in their ground state. That case is more complicated as one should really use ﬂexible constraints to separate the quantum degrees of freedom rather than holonomic constraints (see page (v).)

17.10 The generalized equipartition theorem For several applications it is useful to know the correlations between the ﬂuctuations of coordinates and momenta in equilibrium systems. Statements like “the average kinetic energy equals kB T per degree of freedom” (the equipartition theorem) or “velocities of diﬀerent particles are not correlated”

17.10 The generalized equipartition theorem

503

or the virial theorem itself, are special cases of the powerful generalized equipartition theorem, which states that ∂H = kB T δij , zi (17.203) ∂zj where zi , i = 1, . . . , 2n, stands, as usual, for any of the canonical variables {q, p}. We prove this theorem for bounded systems in the canonical ensemble, but it is also valid for other ensembles. Huang (1987) gives a proof for the microcanonical ensemble. Proof We wish to prove

∂H zi ∂z exp[−βH] dz j = kB T δij . exp[−βH] dz

Consider the following equality −βzi

∂ −βH ∂H −βH ∂e−βH e = zi = zi e − δij e−βH . ∂zj ∂zj ∂zj

Integrating over phase space dz, the ﬁrst term on the r.h.s. drops out after partial integration, as the integrand vanishes at the boundaries. Hence −βzi

∂H = −δij , ∂zj

which is what we wish to prove. ˙ i ; zj = pj we ﬁnd Applying this theorem to zi = pi = (Mq) ˙ i q˙j = δij , (Mq)

(17.204)

q˙ q˙ T = M−1 kB T.

(17.205)

or, in matrix notation:

This is the classical equipartition: for cartesian coordinates (diagonal mass tensor) the average kinetic energy per degree of freedom equals 12 kB T and velocities of diﬀerent particles are uncorrelated. For generalized coordinates the kinetic energy per degree of freedom 12 pi q˙i still averages to 12 kB T , and the velocity of one degree of freedom is uncorrelated with the momentum of any other degree of freedom. Another interesting special case is obtained when we apply the theorem to zi = qi ; zj = qj : qi p˙j = −qi Fj = kB T δij .

(17.206)

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Review of statistical mechanics

For j = i this recovers the virial theorem (see (17.132) on page 485): 1 1 qi Fi = nkB T = Ekin . Ξ=− 2 2 n

(17.207)

i=1

Including the cases j = i means that the virial tensor is diagonal in equilibrium, and the average pressure is isotropic.

Exercises 17.1

17.2 17.3

17.4

17.5

17.6

17.7

17.8

17.9

Show for the canonical ensemble, where Q is a function of V and β, that U = −∂ ln Q/∂β. Show that this equation is equivalent to the Gibbs–Helmholtz equation, (16.32). If the pressure p is deﬁned as the ensemble average of ∂Ei /∂V , then show that for the canonical ensemble p = β −1 ∂ ln Q/∂V . Derive (17.63) by considering how many points there are in k-space in a spherical shell between k and k+dk. Transform this to a function of ε. Derive the quantum expressions for energy, entropy and heat capacity of the harmonic oscillator. Plot the heat capacity for the quantum and classical case. Show that the quantum correction to the free energy of a harmonic oscillator (17.109) equals the ﬁrst term in the expansion of the exact Aqu − Acl in powers of ξ. See (17.133). Show that S r dS = 3V , with the integral taken over a closed surface enclosing a volume V . Transform from a surface to a volume integral over the divergence of r. Carry out the partial diﬀerentiation of A(V, T ) = −kB T ln Q with respect to volume to obtain the isotropic form of (17.138). Assume a cubic L × L × L lattice and use scaled coordinates r/L. Prove that the rotational kinetic energy of a harmonic homonuclear 2 ) equals the contribution to the virialof diatomic molecule (2× 12 mvrot 2 /( 1 d)). the centripetal harmonic force (2 × mvrot 2 Compute the matrix Z for a triatomic molecule with constrained bond lengths.

18 Linear response theory

18.1 Introduction There are many cases of interest where the relevant question we wish to answer by simulation is “what is the response of the (complex) system to an external disturbance?” Such responses can be related to experimental results and thus be used not only to predict material properties, but also to validate the simulation model. Responses can either be static, after a prolonged constant external disturbance that drives the system into a nonequilibrium steady state, or dynamic, as a reaction to a time-dependent external disturbance. Examples of the former are transport properties such as the heat ﬂow resulting from an imposed constant temperature gradient, or the stress (momentum ﬂow) resulting from an imposed velocity gradient. Examples of the latter are the optical response to a speciﬁc sequence of laser pulses, or the time-dependent induced polarization or absorption following the application of a time-dependent external electric ﬁeld. In general, responses can be expected to relate in a non-linear fashion to the applied disturbance. For example, the dielectric response (i.e., the polarization) of a dipolar ﬂuid to an external electric ﬁeld will level oﬀ at high ﬁeld strengths when the dipoles tend to orient fully in the electric ﬁeld. The optical response to two laser pulses, 100 fs apart, will not equal the sum of the responses to each of the pulses separately. In such cases there will not be much choice other than mimicking the external disturbance in the simulated system and “observing” the response. For time-dependent responses such simulations should be repeated with an ensemble of diﬀerent starting conﬁgurations, chosen from an equilibrium distribution, in order to obtain statistically signiﬁcant results that can be compared to experiment. In this chapter we will concentrate on the very important class of linear responses with the property that the response to the sum of two disturbances 505

506

Linear response theory

equals the sum of responses to each of the disturbances. To this class belong all responses to small disturbances in the linear regime; these are then proportional to the amplitude of the disturbance. The proportionality constant determines transport coeﬃcients such as viscosity, thermal conductivity and diﬀusion constant, but also dielectric constant, refractive index, conductivity and optical absorption. Since the decay of a small perturbation, caused by an external disturbance, is governed by the same equations of motion that determine the thermal ﬂuctuations in the equilibrium system, there is a relation between the decay function of an observable of the system after perturbation and the time-correlation function of spontaneous ﬂuctuations of a related variable. In the next sections we shall elaborate on these relations. In Section 18.2 the general relations between an external disturbance and the resulting linear response will be considered both in the time and frequency domain, without reference to the processes in the system that cause the response. In Section 18.3 the relation between response functions and the time correlation function of spontaneously ﬂuctuating quantities will be considered for a classical system of particles that interact according to Hamilton’s equations of motion.

18.2 Linear response relations In this section we consider our system as a black box, responding to a disturbance X(t) with a response Y (t) (Fig. 18.1). The disturbance is an external force or ﬁeld acting on the system, such as an electric ﬁeld E(t), and the response is an observable of the system, for example, a current density j(t) resulting from the disturbance E(t). Both X and Y may be vectorial quantities, in which case their relations are speciﬁed by tensors, but for simplicity of notation we shall stick to scalars here. Exactly how the interactions between particles lead to a speciﬁc response does not concern us in this section. The only principles we assume the system to obey are: (i) (causality) the response never precedes its cause; (ii) (relaxation) the response will, after termination of the disturbance, in due time return to its equilibrium value. Without loss of generality we will assume that the equilibrium value of Y is zero. So, when X = 0, Y (t) will decay to zero. A crucial role in the description of linear responses is played by the delta-

18.2 Linear response relations X(t) -

system

X(t) = X0∗ δ(t)

507

- Y (t) Y (t) = X0∗ Φ(t)

δ-response 0

0

X(t) = X0 H(t)

Y (t) = X0

t 0

Φ(τ ) dτ

step-response 0

0

X(t) = X0 [1 − H(t)]

Y (t) = X0

∞ t

Φ(τ ) dτ

steady-state decay 0

0

Figure 18.1 Black-box response to a small perturbation. Responses to a deltadisturbance, to a step disturbance (Heaviside function) and following a terminated steady state disturbance are sketched.

response Φ(t) (Fig. 18.1): if X(t) = X0∗ δ(t), then Y (t) = X0∗ Φ(t),

(18.1)

where X0∗ is the amplitude of the driving disturbance, taken small enough for the system response to remain linear. Note that we indicate this δ-function amplitude with a star, to remind us that X0∗ is not just a special value of X, but that it is a diﬀerent kind of quantity with a dimension equal to the dimension of X, multiplied by time. Our two principles assure that Φ(t) = 0 for t < 0, lim Φ(t) = 0.

t→∞

(18.2) (18.3)

The shape of Φ(t) is determined by the interactions between particles that govern the time evolution of Y . We note that Φ(t) is the result of a macroscopic experiment and therefore is an ensemble average: Φ(t) is the average result of delta-disturbances applied to many conﬁgurations that are representative for an equilibrium distribution of the system. Because of the linearity of the response, once we know Φ(t), we know the response to an arbitrary disturbance X(t), as the latter can be reconstructed

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Linear response theory

from a sequence of δ-pulses. So the response to X(t) is given by ∞ X(t − τ )Φ(τ ) dτ. Y (t) =

(18.4)

0

A special case is the response to a step disturbance, which is zero for t < 0 and constant for t ≥ 0 (i.e., the disturbance is proportional to the Heaviside function H(t) which is deﬁned as 0 for t < 0 and 1 for t ≥ 0). The response is then proportional to the integral of the δ-response function. Similarly, the response after suddenly switching-oﬀ a constant disturbance (leaving the system to relax from a steady state), is given by the integral from t to ∞ of the δ-response function. See Fig. 18.1. Another special case is a periodic disturbance + , X(t) = X(ω)eiωt , (18.5) which is a cosine function if X( ω) is real, and a sine function if X( ω) is purely imaginary. Inserting this into (18.4) we obtain a response in the frequency domain, equal to the one-sided Fourier transform of the delta-response:

∞ iωt −iωτ Y (t) = X( ω)e Φ(τ )e dτ . (18.6) 0

Writing simply X(t) = X(ω)eiωt and Y (t) = Y (ω)eiωt ,

(18.7)

with the understanding that the observables are the real part of those complex quantities, (18.6) can be rewritten in terms of the complex frequency response χ(ω): Y (ω) = χ(ω)X(ω), with

∞

χ(ω) =

Φ(τ )e−iωτ dτ.

(18.8)

(18.9)

0

The frequency response χ(ω) is a generalized susceptibility, indicating how Y responds to X. It can be split into a real and imaginary part: χ(ω) = χ (ω) − iχ (ω) ∞ Φ(τ ) cos ωτ dτ χ (ω) = 0 ∞ Φ(τ ) sin ωτ dτ χ (ω) = 0

(18.10) (18.11) (18.12)

18.2 Linear response relations

509

Note that the zero-frequency value of χ (which is real) equals the steadystate response to a constant disturbance: ∞ Φ(τ ) dτ, (18.13) χ(0) = 0

X(t) = X0 H(t) → Y (∞) = X0 χ(0).

(18.14)

For the case that X is an electric ﬁeld and Y a current density, χ is the speciﬁc conductance σ. Its real part determines the current component in phase with the periodic ﬁeld (which is dissipative), while its imaginary part is the current component 90◦ out of phase with the ﬁeld. The latter component does not involve energy absorption from the ﬁeld and is usually indicated with the term dispersion. Instead of the current density, we could also consider the induced dipole density P as the response; P is related to j since j = dP/dt. With Y = P , χ becomes ε0 times the electrical susceptibility χe (see Chapter 13) and σ becomes indistinguishable from iωε0 χe . Thus the real part of the electrical susceptibility (or the dielectric constant, or the square root of the refractive index) corresponds to the non-dissipative dispersion, while the imaginary part is a dissipative absorption.

The Kramers–Kronig relations There are interesting relations between the real and imaginary parts of a frequency response function χ(ω), resulting from the causality principle. These are the famous Kramers–Kronig relations:1 2 ∞ ω χ (ω ) χ (ω) = dω , (18.15) π 0 ω 2 − ω 2 2 ∞ ωχ (ω ) dω . (18.16) χ (ω) = π 0 ω 2 − ω 2 Since the integrands diverge when ω approaches ω, the integrals are not well-deﬁned. They must be interpreted as the principal value: exclude an interval ω − ε to ω + ε from the integration and then take the limit ε → 0. The relations arise from the fact that both χ and χ follow from the same delta-response function through equations (18.11) and (18.12). The proof of (18.16) is given below. The reader is challenged to prove (18.15). Proof Start from (18.9). This is a one-sided Fourier (or Fourier–Laplace) 1

These relations were ﬁrst formulated by Kronig (1926) and Kramers (1927) and can be found in many textbooks, e.g., McQuarrie (1976).

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Linear response theory

transform, but taking into account that Φ(τ ) = 0 for τ < 0, the integral can be taken from −∞ instead of zero. Thus we obtain ∞ χ(ω) = χ − iχ = Φ(τ )e−iωτ dτ, −∞

with inverse transform 1 Φ(τ ) = 2π

∞

χ(ω)eiωτ dω.

−∞

Using the symmetry properties of χ and χ (see (18.11) and (18.12)): χ (−ω) = χ (ω) and χ (−ω) = −χ (ω), we can write 1 Φ(τ ) = π

0

∞

1 χ (ω) cos ωτ dω + π

∞

χ (ω) sin ωτ dω.

0

Now, using the fact that Φ(τ ) = 0 for τ < 0, we see that both integrals on the r.h.s. must be equal (for negative τ the last integral changes sign and the total must vanish). Therefore ∞ ∞ π χ (ω) cos ωτ dω = χ (ω) sin ωτ dω = Φ(τ ). 2 0 0 The equality is a direct result of the causality principle. Now insert the ﬁrst expression for Φ(τ ) into (18.12) and obtain ∞ 2 ∞ dτ sin ωτ dω χ (ω ) cos ω τ χ (ω) = π 0 0 ∞ 2 ∞ dω χ (ω ) dτ sin ωτ cos ω τ. = π 0 0 Note that we have changed the integration variable to ω to avoid confusion with the independent variable ω. After rewriting the product of sin and cos as a sum of sines, the last integral can be evaluated to 1 1 ω 1 + = 2 2 ω+ω ω−ω ω − ω 2 Here the primitive function has been assumed to vanish at the limit τ → ∞ because of the inﬁnitely rapid oscillation of the cosines (if this does not satisfy you, multiply the integrand by a damping function exp(−ετ ), evaluate the integral and then take the limit ε → 0). Equation 18.16 results.

18.3 Relation to time correlation functions

511

18.3 Relation to time correlation functions In this section we consider our black box as a gray box containing a system of mutually interacting particles. We assume that the system obeys the classical Hamilton equations and is – in the absence of a disturbance – in an equilibrium state. In the linear regime the perturbation is so small that the system deviates only slightly from equilibrium. When after a deltadisturbance some observable Y deviates slightly from its equilibrium value, it will relax back to the equilibrium value through the same intra-system interactions that cause spontaneous thermal ﬂuctuations of Y to relax. Therefore we expect that the time course of the relaxation of Y after a small perturbation (i.e., the time course of Φ(t)) is directly related to the time correlation function of Y . Let us be more precise. Consider the time correlation function of Y : Y (0)Y (t). The triangular brackets stand for an ensemble average. For a system in equilibrium, presumed to be ergodic, the ensemble average is also a time average over the initial time, here taken as the origin of the time scale. If the probability of Y in the equilibrium ensemble is indicated by Peq (Y ), we can write (18.17) Y (0)Y (t) = Y0 Y (t)|Y (0) = Y0 Peq (Y0 ) dY0 , where Y (t)|Y (0) = Y0 is the conditional ensemble-averaged value of Y at time t, given the occurrence of Y0 at time 0. But that is exactly the response function after an initial disturbance of Y to Y0 : Y (t)|Y (0) = Y0 = Y0

Φ(t) . Φ(0)

(18.18)

Here Φ(0) is introduced to normalize Φ. Inserting this into (18.17), we arrive at the equality: Y (0)Y (t) Φ(t) = . (18.19) Φ(0) Y 2 This relation is often called the ﬁrst ﬂuctuation–dissipation theorem (Kubo, 1966); see for a further discussion page 258. The response Y (0) after a delta-disturbance X(t) = X0∗ δ(t) equals X0∗ Φ(0) (see (18.1)). Therefore: Φ(0) = Y (0)/X0∗ .

(18.20)

This ratio can normally be computed without knowledge of the details of the intra-system interactions, as the latter have no time to develop during a delta-disturbance. The value of Y 2 follows from statistical mechanical

512

Linear response theory

considerations and appears to be related to the delta-response. This relation between Y (0) and Y 2 , which we shall now develop, forms the basis of the Green-Kubo formula that relates the integral of time correlation functions to transport coeﬃcients. Following a delta-disturbance X0∗ δ(t), the point in phase space z (we use symplectic notation, see Section 17.8 on page 492) will shift to z + Δz. The shift Δz is proportional to X0∗ . For example, when the perturbation is an electric ﬁeld E ∗0 δ(t), the ith particle with (partial) charge qi will be subjected to a force qi E ∗0 δ(t), leading to a shift in momentum Δpi = qi E ∗0 . The phase-point shift leads to a ﬁrst-order shift in the response function Y , which is simply a property of the system determined by the point in phase space: 2n ∂Y (z) Δzi . (18.21) ΔY = ∂zi i=1

The delta-response Y (0) =

X0∗ Φ(0)

Y (0) = ΔY =

is the ensemble average of ΔY :

dz e−βH(z)

2n i=1

Δzi

∂Y (z) . ∂zi

(18.22)

By partial integration it can been shown (see Exercise 18.4) that Y (0) = βY ΔH,

(18.23)

where ΔH =

2n i=1

Δzi

∂H(z) . ∂zi

(18.24)

Combining (18.19), (18.20) and (18.24), we ﬁnd a relation between the deltaresponse Φ(t) and the autocorrelation function of Y : Φ(t) =

β Y ΔH Y (0)Y (t). X0∗ Y 2

(18.25)

This relation is only simple if Y ΔH is proportional to Y 2 . This is the case if ΔH ∝ X0∗ Y,

(18.26)

imposing certain conditions on X and Y . An example will clarify these conditions. Consider a system of (partially) charged particles with volume V, subjected to a homogeneous electric ﬁeld E(t) = E0∗ δ(t). For simplicity we consider one dimension here; the extension to a 3D vector is trivial. Each particle with charge qi will experience a

18.3 Relation to time correlation functions

513

force qi E(t) and will be accelerated during the delta disturbance. After the disturbance, the ith particle will have changed its velocity with Δvi = (qi /mi )E0∗ . The total Hamiltonian will change as a result of the change in kinetic energy: ΔH = Δ

1 i

2

mi vi2 =

mi vi Δvi = E0∗

i

q i vi .

(18.27)

i

Thus ΔH is proportional to the current density j: j=

1 q i vi . V

(18.28)

i

So, if we take Y = j, then ΔH = E0∗ V j and (18.25) becomes Φ(t) =

V j(0)j(t). kB T

(18.29)

Note that we have considered one dimension and thus j is the current density in one direction. In general j is a vector and Φ is a tensor; the relation then is V Φαβ (t) = jα (0)jβ (t). (18.30) kB T In isotropic materials Φ will be a diagonal tensor and Φ = by Φ(t) =

V j(0) · j(t). 3kB T

1 3

tr Φ is given

(18.31)

Equation (18.31) relates the correlation function of the equilibrium current density ﬂuctuation with the response function of the speciﬁc conductance σ, which is the ratio between current density and electric ﬁeld: j = σE.

(18.32)

Using (18.9), we can express the frequency-dependent speciﬁc conductance in terms of current density ﬂuctuations: ∞ V σ(ω) = j(0) · j(τ )e−iωτ dτ, (18.33) 3kB T 0 with the special case for ω = 0: V σ0 = 3kB T

0

∞

j(0) · j(τ ) dτ.

(18.34)

514

Linear response theory

Note that in these equations the average product of two currentdensities, multiplied by the volume, occurs: 1 qi vi (0) qi vi (t) , (18.35) V j(o)j(t) = V i

i

which is indeed a statistically stationary quantity when the total volume of the system is much larger than the local volume over which the velocities are correlated. Equation (18.31) is an example of a Kubo formula (Kubo et al., 1985, p. 155), relating a time correlation function to a response function. Equation (18.34) is an example of a Green–Kubo formula (Green, 1954; Kubo, 1957), relating the integral of a time correlation function to a transport coeﬃcient. There are many such equations for diﬀerent transport properties. In the conjugate disturbance E and current density j, which obey the simple relation (18.26), we recognize the generalized force and ﬂux of the thermodynamics of irreversible processes (see (16.98) in Section 16.10 on page 446). The product of force and ﬂux is an energy dissipation that leads to an irreversible entropy production. Kubo relations also exist for other force–ﬂux pairs that are similarly conjugated. In this section we have considered the speciﬁc conductance as example of the Kubo and Green–Kubo formula. In the following sections other transport properties will be considered.

18.3.1 Dielectric properties When the material is non-conducting and does not contain free charge carri ers, the current density j = (1/V ) qi vi is caused by the time derivative P˙ of the dipole density P = (1/V ) qi xi . In this case there is no steady-state current and the zero-frequency conductivity vanishes. But we can connect to the conductivity case by realizing that the following relations exist between time correlation functions of j = P˙ and P : d P (0)P (t) = P (0)P˙ (t) = −P˙ (0)P (t), (18.36) dt d2 P (0)P (t) = P (0)P¨ (t) = −P˙ (0)P˙ (t) = P¨ (0)P (t). (18.37) dt2 These relations are easily derived when we realize that in an equilibrium system the time axis in correlation functions may be shifted: A(0)B(t) =

18.3 Relation to time correlation functions

515

A(−t)B(0). One of the relations in (18.37) is particularly useful: d2 P (0)P (t) = −j(0)j(t), dt2

(18.38)

as it allows us to translate the current ﬂuctuations into polarization ﬂuctuations. The equivalence of the Kubo formula (18.31) for the polarization response ΦP , which is the integral of the current density response Φ(t), is: ΦP (t) = −

V d P (0) · P (t). 3kB T dt

(18.39)

Realizing that P (ω) = ε0 [εr (ω) − 1]E(ω) (see Section 13.2 on page 336), we ﬁnd for the frequency-dependent dielectric constant: ∞ d V P (0) · P (τ ) e−iωτ dτ. (18.40) εr (ω) = 1 − 3kB T 0 dτ At zero frequency the static dielectric constant is obtained: εr (0) = 1 +

V P 2 . 3kB T

(18.41)

In simulations one monitors the total dipole moment M = i μi = V P and computes (1/V )M (0) · M (t). This is generally not a trivial calculation because M is a single quantity that ﬂuctuates slowly and long simulations are needed to obtain accurate converged values for the correlation function. Matters are somewhat more complicated than sketched above.2 In fact, (18.40) and (18.41) can only be trusted for very dilute systems in which mutual dipole interactions can be ignored. The reason is that the local electric ﬁeld, to which the molecules respond, includes the ﬁelds due to other dipoles and the reaction ﬁeld introduced by boundary conditions. Without derivation we give the correct result (Neumann and Steinhauser, 1983) for the relation between the frequency-dependent dielectric constant and the correlation function of the total dipole moment M (t) for the case that a reaction ﬁeld is employed in the simulation (see Section 6.3.5 on page 164 and (13.82) on page 347 for a description of reaction ﬁelds): ∞ d 1 2εRF + 1 [εr (ω) − 1] − M (0) · M (t) e−iωt dt , = 2εRF + εr (ω) 3ε0 V kB T 0 dt (18.42) 2

The theory relating dipole ﬂuctuations with dielectric constants goes back to Kirkwood (1939). The theory for deriving the dielectric constant from dipole ﬂuctuations in simulations with various boundary conditions is most clearly given by Neumann (1983), with extension to the frequency-dependent case by Neumann and Steinhauser (1983). The theory was tested on a Stockmayer ﬂuid (Lennard–Jones particles with dipole moments) by Neumann et al. (1984).

516

Linear response theory

− f Δ xΔ z

vx(y)

y Δz

x f Δ xΔ z

z Δx

Figure 18.2 Two planes, moving relative to each other in a ﬂuid, experience a viscous drag force proportional to the velocity gradient.

where εRF is the relative dielectric constant used for the reaction ﬁeld. For the static dielectric constant εs = εr (0) it follows that (εs − 1)

1 2εRF + 1 = M 2 . 2εRF + εr (ω) 3ε0 V kB T

(18.43)

Equations (18.42) and (18.43) are implicit equations for εr ; they reduce to the simpler (18.40) and (18.41) when εRF = ∞, i.e., for conducting boundary conditions. These are also valid for the use of complete lattice sums with dipole correction (see the discussion of tin-foil boundary conditions on page 373). For simulations with εRF = 1, i.e., using a cutoﬀ radius for Coulomb interactions, the dipole ﬂuctuations are quenched and become rather insensitive to the the value of the dielectric constant. Such simulations are therefore unsuitable to derive dielectric properties.

18.3.2 Viscosity Consider an isotropic ﬂuid between two plates (each in the xz-plane and separated in the y-direction) that move with respect to each other in the xdirection (Fig. 18.2), causing a laminar ﬂow with velocity gradient dvx /dy.

18.3 Relation to time correlation functions

517

On each plate the ﬂuid will exert a drag force f per unit surface of the xzplane, proportional to the velocity gradient. The proportionality constant is the viscosity coeﬃcient η: f =η

∂vx . ∂y

(18.44)

According to (17.113) on page 481 (see also Fig. 17.7), the force per unit surface in a continuum is determined by the stress tensor σ: dF = σ · dS.

(18.45)

For example, on the lower plane in Fig. 18.2, with dS in the y-direction, Fx = σxy Sy = σxy ΔxΔz. This phenomenological deﬁnition agrees with the deﬁnition of η as given in the derivation of the Navier–Stokes equation in Section 9.2, where η connects the oﬀ-diagonal elements of the stress tensor with the velocity gradient as follows (see (9.11) and (9.12)): ∂uα ∂uβ (α = β). (18.46) + σαβ = η ∂xβ ∂xα This is the basic equation deﬁning η. How the stress tensor can be determined from simulations is explained in Section 17.7.2 on page 484. The oﬀ-diagonal elements of the average stress tensor over a given volume V are equal to the negative oﬀ-diagonal elements of the pressure tensor,3 which is “measured” by the virial: 1 1 Fiα xiβ = − σαβ (r) d3 r, α = β. (18.47) Pαβ = V V V i

The (Green–)Kubo relations for the viscosity coeﬃcient can be found by following the standard series of steps. We consider the xy-component of σ without loss of generality. (i) Apply a delta-disturbance g ∗ δ(t) to ∂ux /∂y by imposing an additional velocity vix = g ∗ yi δ(t) to each particle in the considered volume. After the delta pulse the ith particle is displaced in the x-direction by Δxi = g ∗ yi . (ii) Compute ΔH as a result of the disturbance, according to (18.24) and using (18.47): ∂H Δxi = −g ∗ Fxi yi = g ∗ V σxy . (18.48) ΔH = ∂xi i

3

i

The pressure tensor also contains a momentum transfer part due to particle velocities, but that part is diagonal.

518

Linear response theory

(iii) Deﬁne the response Y such that Y is proportional to ΔH. This is fulﬁlled for 1 Fxi yi , (18.49) Y = σxy = − V i

g∗ V

Y. for which ΔH = (iv) Find the δ-response function Φ(t) for Y from (18.25): Φ(t) = The end result is V η(ω) = kB T

∞

V σxy (0)σxy (t). kB T

σxy (0)σxy (τ )e−iωτ dτ,

(18.50)

(18.51)

0

with the Kubo–Green relation for ω = 0: ∞ V σxy (0)σxy (τ ) dτ. η0 = kB T 0

(18.52)

In an isotropic ﬂuid all six oﬀ-diagonal elements have the same correlation function and one can best use the average of the correlation functions of all oﬀ-diagonal elements of σ. The determination of viscosity through the Green-Kubo relation requires accurate determination of the (integral) of the correlation function of a heavily ﬂuctuating quantity. Hess (2002a,b) concluded that it is more eﬃcient to use non-equilibrium molecular dynamics (NEMD) to determine viscosity coeﬃcients (see Section 18.5).

18.4 The Einstein relation All Green-Kubo relations contain the integral of an autocorrelation function of a ﬂuctuating observable f (t): ∞ f (0)f (τ ) dτ. (18.53) 0

Numerical evaluations of such integrals are diﬃcult if no knowledge on the analytical form of the tail of the correlation function is available. The statistics on the tail and – consequently – on the integral, is often poor. An alternative is to monitor not f (t), but its integral F (t): t F (t) = f (t ) dt , (18.54) 0

18.5 Non-equilibrium molecular dynamics

519

and observe the behavior of F 2 (t) for large t. The following Einstein relation is valid: ∞ d 2 f (0)f (τ ) dτ. (18.55) lim F (t) = 2 t→∞ dt 0 This means that F 2 (t), plotted versus t, should approach a straight line. A common application is the determination of the single-particle diﬀusion constant ∞ v(0)v(τ ) dτ, (18.56) D= 0

by observing the mean-squared displacement x2 (t), which approaches 2Dt for times much longer than the correlation time of the velocity. Proof We prove (18.55). Consider d 2 F (t) = 2F (t)f (t) = 2 dt By substituting τ = t −

t

t

f (t )f (t) dt .

0

the right-hand side rewrites to t 2 f (t − τ )f (t) dτ. 0

Since the ensemble average does not depend on the time origin, the integrand is a function of τ only and is equal to the autocorrelation function of f (t). Obviously, the limit for t → ∞ yields (18.55). 18.5 Non-equilibrium molecular dynamics When (small) external “forces” are artiﬁcially exerted on the particles in a molecular-dynamics simulation, the system is brought (slightly) out of equilibrium. Following an initial relaxation the system will reach a steady state in which the response to the disturbance can be measured. In such nonequilibrium molecular dynamics (NEMD) methods the “forces” are chosen to represent the gradient that is appropriate for the transport property of interest. When the system under study is periodic, it is consistent to apply a gradient with the same periodicity. This implies that only spatial Fourier components at wave vectors which are integer multiples of the reciprocal basic vectors (2π/lx , 2π/ly , 2π/lz ) can be applied. This limitation, of course, is a consequence of periodicity and the long-wavelength limit must be obtained by extrapolation of the observed box-size dependence. We now consider a few examples (Berendsen, 1991b).

520

Linear response theory

18.5.1 Viscosity In order to measure viscosity, we wish to impose a sinusoidal shear rate over the system, i.e., we wish to exert an acceleration on each particle. This is accomplished by adding at every time step Δt a velocity increment Δvx to every particle Δvix = AΔt cos kyi .

(18.57)

Here, A is a (small) amplitude of the acceleration and k = 2π/ly is the smallest wave vector ﬁtting in the box. Any multiple of the smallest wave vector can also be used. When m is the mass of each particle and ρ is the number density, the external force per unit volume will be: fxext (y) = mρA cos ky.

(18.58)

Since there is no pressure gradient in the x-direction, the system will react according to the Navier–Stokes equation: mρ

∂ 2 ux ∂ux =η + mρA cos ky. ∂t ∂y 2

(18.59)

The steady-state solution, which is approached exponentially with a time constant equal to mρ/ηk2 (Hess, 2002b), is ux (y) =

mρ A cos ky. ηk2

(18.60)

Thus, the viscosity coeﬃcient η is found by monitoring the gradient in the y-direction of the velocities vx of the particles. Velocity gradients in non-equilibrium molecular dynamics simulations are determined by a leastsquares ﬁt of the gradient to the particle velocities. A periodic gradient can be measured by Fourier analysis of the velocity distribution.

18.5.2 Diﬀusion Self-diﬀusion coeﬃcients can easily be measured from an equilibrium simulation by monitoring the mean-square displacement of the particle as a function of time and applying the Einstein relation. The diﬀusion coeﬃcient measured this way corresponds to the special case of a tracer diﬀusion coeﬃcient of a single tagged particle that has the same interactions as the other particles. In general, the tagged particles can be of a diﬀerent type and can occur in any mole fraction in the system. If not dilute, the diﬀusion ﬂux is inﬂuenced by the hydrodynamic interaction between the moving particles.

18.5 Non-equilibrium molecular dynamics

521

Consider a binary mixture of two particle types 1 and 2, with mole fractions x1 = x and x2 = 1 − x. Assume that the mixture behaves ideally: μi = μ0i + RT ln xi .

(18.61)

Now we can derive the following equation: u1 − u2 = −

D ∇x. x(1 − x)

(18.62)

In an NEMD simulation we apply two accelerations a1 and a2 to each of the particles of type 1 and 2, respectively. This is done by increasing the velocities (in a given direction) every step by a1 Δt for species 1 and by a2 Δt for species 2. The total force on the system must be kept zero in order to avoid acceleration of the center of mass: M1 xa1 + M2 (1 − x)a2 = 0,

(18.63)

where M1 and M2 are the molar masses of species 1 and 2, respectively. The balance between driving force and frictional force is reached when M2 a2 = −

RT x(u1 − u2 ), D

(18.64)

or, equivalently, M1 a1 =

RT (1 − x)(u1 − u2 ). D

(18.65)

When u1 − u2 is monitored after a steady state has been reached, the diffusion coeﬃcient is easily found from either of these equations. The steady state is reached exponentially with a time constant equal to M D/RT . The amplitudes a1 and a2 should be chosen large enough for a measurable eﬀect and small enough for a negligible disturbance of the system; in practice the imposed velocity diﬀerences should not exceed about 10% of the thermal velocities.

18.5.3 Thermal conductivity The thermal conductivity coeﬃcient Λ can be measured by imposing a thermal ﬂux Jq with the system’s periodicity: Jq (y) = A cos ky.

(18.66)

This is accomplished by scaling the velocities of all particles every time step by a factor λ. The kinetic energy Ekin per unit volume changes by a factor

522

Linear response theory

λ2 , such that ΔEkin (λ2 − 1) 3 (18.67) = ρkB T. Δt Δt 2 The external heat ﬂow causes a temperature change; the temperature T (y) will obey the following equation: Jq =

ρcv

∂2T ∂T = Jq + Λ 2 . ∂t ∂y

(18.68)

The steady-state solution then is A cos ky. (18.69) Λk 2 Thus, by monitoring the Fourier coeﬃcient of the temperature at wave vector k in the y-direction, the thermal conductivity coeﬃcient Λ is found. Also here, the amplitude of the heat ﬂux should be chosen large enough for a measurable eﬀect and small enough for a negligible disturbance. In practice a temperature amplitude of 10 K is appropriate. T (y) = T0 +

Exercises 18.1

18.2 18.3 18.4

18.5

18.6

When the delta response of a linear system equals an exponential decay with time constant τc , compute the response to a step function and the frequency response (18.9) of this system. Verify the validity of he Kramers–Kronig relations for the frequency response of the previous exercise. Prove (18.15) by following the same reasoning as in the proof given for (18.16). Prove (18.23) by showing through partial integration that ∂Y ∂H −βH(z) e Δzi dz = β e−βH(z) Y Δzi dz. (E18.1) ∂zi ∂zi When the total dipole ﬂuctuation M (0) · M (t) appears to decay exponentially to zero in a simulation with “tin-foil” boundary conditions, how then do the real and imaginary parts of the dielectric constant depend on ω? Consider εr (ω) = εr − iεr . Plot εr versus εr and show that this Cole–Cole plot is a semicircle. Derive (18.62).

19 Splines for everything

19.1 Introduction In numerical simulations one often encounters functions that are only given at discrete points, while values and derivatives are required for other values of the argument. Examples are: (i) the reconstruction of an interaction curve based on discrete points, for example, obtained from extensive quantum calculations; (ii) recovery of function values and derivatives for arbitrary arguments from tabulated values, for example, for potentials and forces in MD simulations; (iii) the estimation of a deﬁnite or indeﬁnite integral of a function based on a discrete set of derivatives, for example, the free energy in thermodynamic integration methods; (iv) the construction of a density distribution from a number of discrete events, for example, a radial distribution function. In all these cases one looks for an interpolation scheme to construct a complete curve from discrete points or nodes. Considerations that inﬂuence the construction process are (i) the smoothness of the curve, (ii) the accuracy of the data points, and (iii) the complexity of the construction process. Smoothness is not a well-deﬁned property, but it has to do with two aspects: the number of continuous derivatives (continuous at the nodes), and the integrated curvature C, which can be deﬁned as the integral over the square of the second derivative over the relevant interval:

b

C[f ] =

{f (x)}2 dx,

a

523

(19.1)

524

Splines for everything

or, in the multidimensional case:

{∇2 f (r)}2 dr.

C[f ] =

(19.2)

V

The term “curvature” is used very loosely here.1 If data points are not inﬁnitely accurate, there is no reason why the curve should go exactly through the data points. Any curve from which the data points deviate in a statistically acceptable manner, is acceptable from the point of view of ﬁtting to the data. In order to choose from the – generally inﬁnite number of – acceptable solutions, one has to apply additional criteria, as compliance with additional theoretical requirements, minimal curvature, minimal complexity of the curve speciﬁcation, or maximum “uncertainty” (“entropy”) from an information-theoretical point of view. Finally, one should always choose the simplest procedure within the range of acceptable methods. The use we wish to make of the constructed function may prescribe the number of derivatives that are continuous at the nodes. For example, in an MD simulation using an algorithm as the Verlet or leap-frog scheme, the ﬁrst error term in the prediction of the coordinate is of the order of (Δt)4 . The accuracy depends on the cancellation of terms in (Δt)3 , which involve the ﬁrst derivatives of the forces. For use in such algorithms we wish the derivative of the force to be continuous, implying that the second derivative of the potential function should be continuous. Thus, if potential and forces are to be derived from tabulated functions, the interpolation procedure should not only yield continuous potentials, but also continuous ﬁrst and second derivatives of the potential. This, in fact, is accomplished by cubic spline interpolation. Using cubic spline interpolation, far less tabulated values are needed for the same accuracy than when a simple linear interpolation scheme would have been used. We ﬁrst restrict our considerations to the one-dimensional case of polynomial splines, which consist of piecewise polynomials for each interval. These local polynomials are by far to be preferred to global polynomial ﬁts, which are ill-behaved and tend to produce oscillating solutions. The polynomial splines cannot be used when the function is not single-valued or when the x-coordinates of the data points cannot be ordered (x0 ≤ x1 ≤ · · · ≤ xn ); one then needs to use parametricn splines, where both x and y (and any 1

It would be better to call the curvature, as deﬁned here, the total energy of curvature. Curvature is deﬁned as the change of tangential angle per unit of length along the curve, which is the inverse of the radius of the circle that ﬁts the local curved line segment. This curvature is a local property, invariant for orientation of the line segment. For a function y(x) the curvature can be expressed as y (1 + y )−3/2 , which can be approximated by y (x). If an elastic rod is bent, the elastic energy per unit of rod length is proportional to the square of the curvature. See Bronstein and Semendjajew (1989) for deﬁnitions of curvature.

19.2 Cubic splines through points

525 n

t

yn n−1t

yi+1 yi

t t

t

t

t

i+1

i

1

y0

t0 x0

hi xi

xi+1

xn

Figure 19.1 Cubic spline interpolation of n + 1 points in n intervals. The i-th interval has a width of hi and runs from xi to xi+1 . First and second derivatives are continuous at all nodes.

further coordinates in higher-dimensional spaces) are polynomial functions of one (or more, in higher-dimensional spaces) parameter(s). To this category belong B-splines and Bezier curves and surfaces. B-splines are treated in Section 19.7; they are often used to construct smooth surfaces and multidimensional interpolations, for example in the smooth-particle mesh-Ewald (SPME) method for computing long-range interactions in periodic systems (see Section 13.10.6 in Chapter 13 on page 373). For Bezier curves we refer to the literature. For many more details, algorithms, programs in C, and variations on this theme, the reader is referred to Engeln-M¨ ullges and Uhlig (1996). Programs including higher-dimensional cases are also given in Sp¨ ath (1973). A detailed textbook is de Boor (1978), while Numerical Recipes (Press et al., 1992) is a practical reference for cubic splines. In the following sections the spline methods are introduced, with emphasis on the very useful cubic splines. A general Python program to compute onedimensional cubic splines is provided in Section 19.6. Section 19.7 treats B-splines.

526

Splines for everything

19.2 Cubic splines through points Consider n + 1 points xi , yi , i = 0, . . . , n (Fig. 19.1). We wish to construct a function f (x) consisting of piece-wise functions fi (x − xi ), i = 0, . . . , n − 1, deﬁned on the interval [0, hi = xi+1 − xi ], with the following properties: (i) (ii) (iii) (iv) (v) (vi)

each fi (ξ) is a polynomial of the third degree; fi (0) = yi , i = 0, . . . , n − 1; fn−1 (hn−1 ) = yn ; fi (hi ) = fi+1 (0), i = 0, . . . , n − 2; (0), i = 0, . . . , n − 2; fi (hi ) = fi+1 fi (hi ) = fi+1 (0), i = 0, . . . , n − 2.

There are n intervals with 4n parameters to describe the n piecewise functions; the properties provide (n+1)+3(n−1) = 4n−2 equations. In order to solve for the unknown parameters, two extra conditions are needed. These are provided by speciﬁcation of two further derivatives at the end nodes, choosing from the ﬁrst, second or third derivatives of either or both end nodes.2 In the case of periodic cubic splines, with period xn − x0 , for which yn = y0 , it is speciﬁed that the ﬁrst and second derivative at the ﬁrst point are equal to the ﬁrst and second derivative at the last point. If it is speciﬁed that the second derivatives at both end points vanish, natural splines result. It is clear that quadratic splines will be obtained when the function and its ﬁrst derivative are continuous (one extra condition being required), and that quartic and higher-order splines can be deﬁned as well. The function in the ith interval is given by f (xi + ξ) = fi (ξ) = fi + gi ξ + pi ξ 2 + qi ξ 3

(0 ≤ ξ ≤ hi ).

(19.3)

Here fi , gi , pi and qi are the four parameters to be determined from the conditions given above. We see immediately that the values of the function and its ﬁrst derivative in point xi are fi = f (xi ) = yi ,

(19.4)

gi = f (xi ) = fi ,

(19.5)

and

respectively, while 2pi is the second derivative at xi and 6qi is the third derivative, which is constant throughout the i-th interval, and discontinuous at the nodes. 2

It is wise to specify one condition at each end node; specifying two conditions at one end node may lead to accumulating errors giving erratic oscillations.

19.2 Cubic splines through points

527

The continuity conditions lead to the following equations: fi (hi ) = fi + gi hi + pi h2 + qi h3 = fi+1 ,

(19.6)

fi (hi ) fi (hi )

(19.7)

= gi + 2pi hi +

3qi h2i

= gi+1 =

= 2pi + 6qi hi = 2pi+1 =

fi+1 ,

fi+1 ,,

(19.8)

valid for i = 0, . . . , n − 2. Four parameters suﬃce for the reconstruction of f (x) in each interval. It is convenient to use the function values and the ﬁrst derivatives in the nodes for reconstruction. Thus, for the i-th interval we obtain by solving pi and qi from (19.6) and (19.7):

1 3(fi+1 − fi ) − 2gi − gi+1 , (19.9) pi = hi hi

1 −2(fi+1 − fi ) + gi + gi+1 . qi = 2 (19.10) hi hi Any value of f (x) within the i-th interval is easily found from (19.3) when both the function and derivative values are given at the nodes of that interval. So the interpolation is practically solved when the ﬁrst derivatives are known, and the task of constructing a cubic spline from a set of data points is reduced to the task of ﬁnding the ﬁrst derivatives in all points (including the end nodes).3 The continuity of the second derivatives is given by (19.8). Written in terms of f and g, this condition gives n − 1 equations for n + 1 unknowns g0 , . . . , gn : 1 1 1 1 gi+1 + gi + 2 + gi+2 hi hi hi+1 hi+1 fi+1 − fi fi+2 − fi+1 , i = 0, . . . , n − 2. (19.11) = 3 + h2i h2i+1 These equations must be augmented by the two additional conditions, and – depending on the exact conditions – will lead to a matrix equation with a symmetric, tridiagonal matrix. For example, if the ﬁrst derivatives g0 and gn at the end points are known, they can be removed from the unknown vector and lead to a matrix equation with g = [g1 , . . . gn−1 ] as the unknown vector: Ag = b, 3

(19.12)

We could have chosen to solve for the second derivatives at all nodes, as is often done in the literature, since the functions in each interval are also easily constructed from knowledge of the values of the function and its second derivatives at the nodes. The computational eﬀort is similar in both cases.

528

⎛

d1 ⎜ s1 ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

Splines for everything

s1 d2 .. .

⎞⎛

s2 .. .. . . .. . dn−2 sn−2 sn−2 dn−1

g1 g2 .. .

⎛

⎞

⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠ gn−2 gn−1

⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟=⎜ ⎟ ⎜ ⎠ ⎝

b1 − g0 /h0 b2 .. . bn−2 bn−1 − gn /hn−1

⎞ ⎟ ⎟ ⎟ ⎟, ⎟ ⎠

where we use the general notation si = h−1 i , i = 1, . . . , n − 1,

(19.13)

and di = 2(si−1 + si ), i = 1, . . . , n − 1 bi =

3s2i−1 (fi

− fi−1 ) +

3s2i (fi+1

(19.14)

− fi ), i = 1, . . . , n − 1.

(19.15)

In case the second derivative is given at an end node, say at x0 : f0 = 2p0 , the extra condition reads h0 p0 = Now g0 cannot be to the matrix: ⎛ d0 s0 ⎜ s0 d1 s1 ⎜ ⎜ s1 d2 ⎜ ⎜ .. ⎜ . ⎜ ⎜ ⎝

3 (f1 − f0 ) − 2g0 − g1 . h0

(19.16)

eliminated and an extra row and column must be added ⎞⎛ s2 .. .. . . .. . dn−2 sn−2 sn−2 dn−1

g0 g1 g2 .. .

⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠ gn−2 gn−1

⎞

⎛

⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟=⎜ ⎟ ⎜ ⎟ ⎜ ⎠ ⎝

b0 b1 b2 .. .

⎞ ⎟ ⎟ ⎟ ⎟ ⎟, ⎟ ⎟ ⎠

bn−2 bn−1 − gn sn−1 (19.17)

with d0 = 2s0 ,

(19.18)

1 b0 = 3(f1 − f0 )s20 − f0 . 2

(19.19)

Again, a symmetric tridiagonal matrix is obtained. Similarly, a given second derivative at the end point xn can be handled., yielding an extra row at the bottom and column at the right with dn = 2sn−1 ,

(19.20)

1 bn = 3(fn − fn−1 )s2n−1 − fn . 2

(19.21)

19.2 Cubic splines through points

529

If the third derivative f0 is speciﬁed (at x0 ), the same matrix is obtained as in the previous case, but with values d0 = s0 ,

(19.22)

1 b0 = 2(f1 − f0 )s20 + f0 h0 . 6

(19.23)

For the third derivative fn speciﬁed at xn , the extra elements are dn = sn−1 ,

(19.24)

1 bn = 2(fn − fn−1 )s2n−1 + fn hn−1 . 6

(19.25)

For periodic splines, f (xn +ξ) = f (x0 +ξ), and the function and its ﬁrst two derivatives are continuous at xn . Thus there are n unknowns g0 , . . . , gn−1 , and the matrix equation now involves a symmetric tridiagonal cyclic matrix: Ag = b, ⎛

d0 ⎜ s0 ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ sn−1

s0 d1 .. .

sn−1 s1 .. .. . . .. . dn−2 sn−2 sn−2 dn−1

(19.26)

⎞⎛ g0 ⎟⎜ ⎟⎜ g1 ⎟⎜ . ⎟⎜ .. ⎟⎜ ⎟⎝ ⎠ gn−2 gn−1

⎞

⎛

b0 b1 .. .

⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟=⎜ ⎟ ⎜ ⎠ ⎝ bn−2 bn−1

⎞ ⎟ ⎟ ⎟ ⎟, ⎟ ⎠

where the matrix elements are given by (19.14), (19.13) and (19.15), with additional elements d0 = 2(s0 + 2sn−1 ), b0 = 3(f1 −

f0 )s20

+ 3(f0 −

(19.27) fn−1 )s2n−1 ,

bn−1 = as in (19.15) with fn = f0 .

(19.28) (19.29)

As the matrices are well-behaved (diagonally dominant, positive deﬁnite), the equations can be simply solved; algorithms and Python programs are given in Section 19.6. Cubic splines have some interesting properties, related to the curvature, as deﬁned in (19.1). We refer to Kreyszig (1993) for the proofs. (i) Of all functions (continuous and with continuous ﬁrst and second derivatives) that pass through n given points, and have given ﬁrst derivatives at the end points, the cubic spline function has the smallest curvature. The cubic spline solution is unique; all other functions

530

Splines for everything

with these properties have a larger curvature. One may say that the cubic spline is the smoothest curve through the points.4 (ii) Of all functions (continuous and with continuous ﬁrst and second derivatives) that pass through n given points, the function with smallest curvature is a natural cubic spline, i.e., with zero second derivatives at both ends. So, if for some good reason, you look for the function with smallest curvature through a number of given points, splines are the functions of choice. Figure 19.2 shows the cubic (periodic) spline solution using as x, y data just the following points, which sample a sine function: 3π π xi = 0, , π, , 2π, 2 2

yi = 0, 1, 0, −1, 0.

The spline function (dotted line) is almost indistinguishable from the sine itself (solid curve), with a largest deviation of about 2%. Its ﬁrst derivatives at x = 0 and x = π diﬀer somewhat from the ideal cosine values. The cubic interpolation using exact derivatives (1, 0, −1, 0, 1) gives a somewhat better ﬁt (dashed curve), but with slight discontinuities of the second derivatives at x = 0 and x = π.

19.3 Fitting splines There is one good reason not to draw spline functions through a number of given points. That is if the points represent inaccurate data. Let us assume that the inaccuracy is a result of statistical random ﬂuctuations.5 The data points yi then are random deviations from values fi = f (xi ) of a function f (x) that we much desire to discover. The value of di = yi − fi is a random sample from a distribution function pi (di ) of the random variable. That is, if the data points are statistically independent; if they are not, the whole set of deviations is a sample from a multivariate probability distribution. We need at least some knowledge of these distribution functions, best obtained from separate observations or simulations: 4

5

This is the rationale for the name spline, borrowed from the name of the thin elastic rods used by construction engineers to ﬁt between pairs of nails on a board, in order to be able to draw smooth outlines for shaping construction parts. The elastic deformation energy in the rod is proportional to the integral of the square of the second derivative, at least for small deviations from linearity. The rod will assume the shape that minimizes its elastic energy. If the ends of the rods are left free, natural splines result. Lasers and automated cutting machines have made hardware splines obsolete. Be aware of, and check for, experimental errors or programming errors, and – using simulations – for insuﬃcient sampling and inadequate equilibration. An observable may appear to be randomly ﬂuctuating, but still only sample a limited domain. This is a problem of ergodicity that cannot be solved by statistical methods alone.

19.3 Fitting splines

531

0.02

1

0.01

0.75

0

0.5

−0.01

0.25

−0.02 1

0

2

3

4

5

6

−0.25 −0.5 −0.75 −1 1

2

3

4

5

6

Figure 19.2 A cubic periodic spline (dotted) ﬁtted through ﬁve points sampling a sine wave (solid curve). Dashed curve: cubic interpolation using function values and ﬁrst derivatives at the sample points. The inset shows the diﬀerences with the sine function.

(i) their expectation (or “expectation value”), deﬁned as the average over the distribution function, must be assumed to be zero. If not, there is a bias in the data that – if known – can be removed. ∞ xpi (x) dx = 0, (19.30) ∞

(ii) their variances σi2 , deﬁned as the expectation of the square of the variable over the unbiased distribution function ∞ 2 x2 pi (x) dx. (19.31) σi = ∞

For our purposes it is suﬃcient to have an estimate of σi . It enables us to determine the sum of weighted residuals, usually indicated by chi-square: χ2 =

n (yi − fi )2 i=0

σi2

.

(19.32)

532

Splines for everything 2 f( χ )

0.15

0.10

5 ν=

10 20

30

50

0.05

0

20

40

60

χ2

Figure 19.3 The chi-square probability distribution for various values of ν, the number of degrees of freedom.

If the weighted deviations (yi −fi )/σi are samples from normal distributions, χ2 will be distributed according to the chi-square distribution ν χ2 f (χ2 |ν) = [2ν/2 Γ( )]−1 (χ2 )ν/2−1 exp(− ) dχ2 , 2 2

(19.33)

as depicted in Fig. 19.3. The parameter ν is the number of degrees of freedom; if the deviations are uncorrelated, ν = n for our purposes. The expectation (average value over the distribution) of χ2 is equal to ν, and for large ν the χ2 -distribution tends to a Gaussian distribution with variance 2ν. From the cumulative χ2 -distribution conﬁdence intervals can be computed: these indicate the range in which, say, 90% of the samples are expected to occur, or, say, the value that will be exceeded in 1% of the cases. Note that the median value for χ2 , expected to be exceeded in 50% of the cases, is close to ν. Chi-square tables can be used to decide what deviation from a ﬁtted curve is considered acceptable. The logical choice is to take the value at the expectation of χ2 , which is the 50% value, for which χ2 = ν is a suﬃcient approximation. If a small value is set, the ﬁtted curve may follow the noisy data too closely; if a large value is allowed, details may be missed that are real. Extensive tables and equations can be found in Beyer (1991) and in Abramowitz and Stegun (1965), as well as in most textbooks on statistics. In Table 19.1 an excerpt is given of values that are not exceeded by χ2 /n in a given fraction of the cases. The 50% value is the most neutral expectation, which statistically will be exceeded in half of the cases. The

19.3 Fitting splines

533

Table 19.1 Values that are not exceeded by χ2 /n in the fraction of cases F mentioned in the header row. Vertical is the number of degrees of freedom ν.

F = 0.01 ν 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 20 25 30 40 50 60 70 80 90 100 ∞

0.000 0.005 0.038 0.074 0.111 0.145 0.177 0.206 0.232 0.256 0.278 0.298 0.316 0.333 0.349 0.413 0.461 0.498 0.554 0.594 0.625 0.649 0.669 0.686 0.701

0.05

0.10

0.50

0.90

0.95

0.99

0.004 0.051 0.117 0.178 0.229 0.273 0.310 0.342 0.369 0.394 0.416 0.436 0.453 0.469 0.484 0.543 0.584 0.616 0.663 0.695 0.720 0.739 0.755 0.768 0.779

0.016 0.105 0.195 0.266 0.322 0.367 0.405 0.436 0.463 0.487 0.507 0.525 0.542 0.556 0.570 0.622 0.659 0.687 0.726 0.754 0.774 0.790 0.803 0.814 0.824

0.455 0.693 0.789 0.839 0.870 0.891 0.907 0.918 0.927 0.934 0.940 0.945 0.949 0.953 0.956 0.967 0.973 0.978 0.983 0.987 0.989 0.990 0.992 0.993 0.993

2.706 2.303 2.084 1.945 1.847 1.774 1.717 1.670 1.632 1.599 1.570 1.546 1.524 1.505 1.487 1.421 1.375 1.342 1.295 1.263 1.240 1.222 1.207 1.195 1.185

3.841 2.996 2.605 2.374 2.214 2.099 2.010 1.938 1.880 1.831 1.789 1.752 1.720 1.692 1.666 1.571 1.506 1.459 1.394 1.350 1.318 1.293 1.273 1.257 1.243

6.635 4.605 3.782 3.319 3.017 2.802 2.639 2.511 2.407 2.321 2.248 2.185 2.130 2.082 2.039 1.878 1.773 1.696 1.592 1.523 1.473 1.435 1.404 1.379 1.358

1 − c 1 −√ b 1−a 1 √1 + a 1 + b 1√+ c a = 1.81/ ν b = 2.32/ ν c = 3.29/ ν

number of degrees of freedom ν is not necessarily equal to the number of data points over which the square deviations are summed; correlation between the points and the “use” of data points to determine parameters in a function to which the data are ﬁtted, will decrease the number of eﬀective degrees of freedom. For example, if a straight line (two parameters) is ﬁtted to two points, there are no deviations and no degrees of freedom; if a straight line is ﬁtted to n independent data points, there are only n − 2

534

Splines for everything

degrees of freedom. If we would duplicate every data point, the values of χ2 and of n would both double, but χ2 /n would not change, and neither would the number of degrees of freedom change. The eﬀective number of degrees of freedom equals the number of data points divided by a correlation length of the data and reduced by the number of (independent) parameters in the ﬁtted function. It is clear that a criterium based on acceptance of a χ2 -value does not suﬃce to choose the parameters of a model. In the context of splines we choose the additional criterium to minimize the curvature of the constructed function. We choose natural splines, as these have the least curvature among all splines through the same points.6 The total curvature C of a cubic spline is given by xn xn n−1 hi 2 xn (f ) dx = f f |x0 − f f dx = − fi fi dξ C = x0

= −6

n−1 i=0

x0

hi

qi

fi dξ = −6

0

n−1

i=0

0

qi (fi+1 − fi ).

(19.34)

i=0

This can be rewritten as

n−1 n (qi − qi−1 )fi − qn−1 fn = 6 δi fi , C = 6 q0 f0 + i=1

(19.35)

i=0

where δi = qi − qi−1 , δ 0 = q0 ,

i = 1, . . . , n − 1, δn = −qn−1 .

(19.36)

In order to minimize C while χ2 is to be constrained at a given value, we use a Lagrange multiplier 7 λ and minimize the function C + λχ2 : Minimize

n 0

δi fi + λ

(yi − fi )2 σi2

.

(19.37)

Here we have absorbed the constant 6 in the Lagrange multiplier. Note that the variables are fi , i = 0, . . . , n, while the δi ’s follow from the spline through fi ; yi and σi are constants. For λ an initial value is chosen, and the 6

7

There are circumstances that external knowledge would imply other boundary conditions than vanishing second derivatives, e.g. in the case of periodic conditions. It is possible to construct programs for other boundary conditions, but in general one may choose some extra points at, or just beyond, the boundaries such that certain prescribed conditions will be fulﬁlled. This is the standard way to optimize a function in the presence of a constraint. For its use see Chapter 17, footnote on page 456.

19.3 Fitting splines

535

procedure is iterated until the desired value of χ2 is obtained. Since χ2 is a statistical quantity, it may well happen that the desired value of χ2 is not even reached with the smoothest possible curve (a straight line with C = 0); in that case the straight line is an acceptable least squares ﬁt to a linear function. In order to solve (19.37), we solve the set of equations obtained by setting the derivatives to all fk equal to zero: δk +

n i=0

fi

yk − fk ∂δi − 2λ = 0, ∂fk σk2

k = 1, . . . , n.

(19.38)

Now, it is easy to show that δ is a homogeneous linear function of f : δ = Mf . The matrix M appears to be symmetric, as can be shown by a straightforward but cumbersome inspection of the matrices M is composed of. We leave the proof to the interested reader. Since ∂δi /∂fk = Mik = Mki , the second term in (19.38) equals Mki fi = δk . Thus λ

yk − fk = δk , σk2

i = 0, . . . , n,

(19.39)

is the condition to be imposed on the spline parameters to obtain the smoothest curve (in terms of least curvature) with a constrained value of χ2 . By varying λ, the smoothest curve that is statistically compatible with the data is obtained. Equation (19.39), together with (19.10), expressing q in f and g, and the continuity conditions that lead to the matrix equation (19.12) or (19.26), yields a matrix equation from which the spline parameters can be solved. The equation is now ﬁve-diagonal but symmetric; complete expressions can be found in Engeln-M¨ ullges and Uhlig (1996), who also give expressions for diﬀerent boundary conditions. A Python program is given in Section 19.6. Figure 19.4 shows ﬁtting splines through 11 points that were generated as normal-distributed random deviations from the parabola y = 0.2x2 . Three ﬁtting splines were generated with diﬀerent overall weight factors for the data points: w = 1000 yields a spline through the points; w = 0.2 yields a value of χ2 /n approximately equal to the mean expectation 1; w = 0.001 yields a least squares ﬁt to an (almost) straight line. Note that the “best” ﬁt (dashed line) is not equivalent to a least squares ﬁt to a polynomial of the second degree. If knowledge is available from other sources about the functional form that is to be expected, a direct least-squares ﬁt to such a functional form is to be preferred above a ﬁtting spline. The latter will always choose the smoothest curve, statistically consistent with the data.

536

Splines for everything

7 original function curv = 237, χ2/n = 0.001 curv = 0.70, χ2/n = 1.02 curv = 0.001, χ2/n = 3.06

6 5 4 3 2 1 0 −1 −2 −6

−4

−2

0

2

4

6

Figure 19.4 Fitting splines approximating noisy data points. Three curves were generated with diﬀerent weighting factors (1000; 0.2; 0.001). The results range from a spline through the points to an almost straight line. The “best” ﬁt has an average chi-square deviation close to 1.

What we are looking for is the smoothest curve consistent not only with the data, but with all the knowledge we may possess about the expected curve.

19.4 Fitting distribution functions Suppose we wish to generate a probability distribution function ρ(x) from a number of statistical samples from that distribution. For example, we wish to generate a radial distribution function between two speciﬁed species of atoms, or an angular distribution of relative orientations of molecular dipoles. If there are suﬃcient independent data, this task simply reduces to constructing a histogram by sorting the data into bins x0 , . . . , xn . The number of samples ni in each bin follows multinomial statistics: when the probability to ﬁnd a sample in the i-th bin is pi , then the joint probability to ﬁnd the distribution n1 , . . . , nn , given the total number of samples N , is given by P (n1 , . . . , nn ) = N ! Πni=1

pni i . ni !

(19.40)

19.4 Fitting distribution functions

537

The (marginal) probability to ﬁnd exactly ni samples in the i-th bin, irrespective of the occupancy of other bins, follows binomial statistics: P (ni |N ) =

N! pni i (1 − pi )N −ni . ni !(N − ni )!

(19.41)

For a suﬃcient number of bins, the probability to ﬁnd a sample in one particular bin becomes small. In the limit of pi → 0 (but keeping the expected number in the bin μi = N pi constant), the binomial distribution becomes a Poisson distribution: P (ni |μi ) =

μni e−μ . ni !

The expected value μi in bin (xi , xi+1 ) equals xi+1 ρ(x) dx. μi = N

(19.42)

(19.43)

xi

The observed number can be written as ni = μi + εi ,

(19.44)

with εi a random variable with average zero and variance μi . Thus the observed number ni has a standard deviation approximately equal to the square root of ni itself. When the number of samples in each bin is large, the histogram itself is a good representation of the required distribution function. When the available number of samples is small, it is not possible to construct a histogram that has both a high resolution and a high precision. Can we construct a smooth distribution function that is compatible with the sampled data? There is one diﬃculty: the number in each bin is not a sample of the distribution function at one point (say, the center of the bin), but rather a sample of the integral of the distribution function over the bin. This diﬃculty is resolved if we start with the cumulative distribution Ni : Ni =

i

nj ,

(19.45)

j=1

which is a sample of (N ×) the cumulative distribution function F (xi ): x ρ(x) dx. (19.46) F (x) = −∞

So we ﬁt a spline to the observed numbers Ni /N , knowing that the function must start at 0 and end at 1. We also know that the derivative of the

538

Splines for everything

% 100 80 60 40 20

-3

-2

-1

0

1

2

3

Figure 19.5 Cumulative distribution of 50 samples taken from a normal distribution.

function, being a density, must always be non-negative. But we must also know the standard deviations of Ni /N . Since i Ni = E[Ni ] + εj , (19.47) j=1

we can approximate by the ﬁrst term in a Taylor expansion Ni 1 1 ≈ F (xi )(1 + εj − εj = F (xi ) + ηi . N E[Ni ] E[N ] i

j=1

n

(19.48)

j=1

The variance of the relative error ηi can be easily computed, using the facts that the variance of εj equals μj (expectation of nj ) and that εi and εj are uncorrelated for i = j. We ﬁnd Ni 1 Ni 2 σi = Fi (1 − Fi ) ≈ 1− . (19.49) N N N The standard deviations are zero at both ends, which √are therefore forced to 0 and 1, and in the middle reaches the value of 0.5/ N . We now consider an example: the construction of the distribution function of 50 samples from a normal distribution. Figure 19.5 shows the cumulative distribution of the data. In Fig. 19.6 the cumulative distribution is put into bins of width 0.5, and a ﬁtting spline is constructed with such a weight that

19.5 Splines for tabulation

1 0.8

539

w = 0.00035 χ2/n = 1.06 curv = 0.101

0.6 0.4 0.2 0 −3

−2

−1

0

1

2

3

Figure 19.6 A ﬁtting spline approximating the cumulative distribution.

0.3

0.2

0.1

0 −3

−2

−1

0

1

2

3

Figure 19.7 Distribution constructed (by a spline) to the ﬁrst derivative of the cumulative ﬁt.

χ2 /n = 1. Finally, Fig. 19.7 shows a spline ﬁt through the derivative values of Fig. 19.6. The distribution is still some way from being Gaussian, but it is the smoothest (cumulative) distribution compatible with the data.

19.5 Splines for tabulation Splines can be used to recover forces and energies from tabulated values. Assume we have a centrosymmetric potential V (r) between a pair of par-

540

Splines for everything

ticles, deﬁned for 0 < r ≤ rc , with rc a cutoﬀ distance beyond which the potential is identically zero. The particles have cartesian coordinates r i and r j ; we use the convention r = r i − r j and r = |r|. The force F = F i on particle i, which is equal and opposite to the force acting on particle j, is given by dV r F (r) = − . (19.50) dr r We wish to tabulate values of V such that: • V and dV /dr are recovered simultaneously from the table; • the derivative of the force is continuous – this is accomplished by using cubic spline interpolation for V ; • the interpolation should be fast – this requires storage of not only the values of the potential, but (at least) also of its derivative, allowing local cubic interpolation; • a search procedure to ﬁnd the table index for an arbitrary argument should be avoided – the table index should follow directly from the argument; • the computationally expensive square root evaluation should be avoided – this means that table indices should be derived from r2 rather than r. These requirements lead to the following procedures. Construction of the table (i) Decide on the number of table entries n + 1 (the table index runs from 0 to n). (ii) Decide on the cutoﬀ radius rc . (iii) Deﬁne a scaled variable x, proportional to r2 , running from 0 to n. x=

nr2 . rc2

(19.51)

The table index i runs from 0 to n. (iv) Construct a table with function values fi and derivatives gi . The values should be the required potential and its derivative. Deﬁning & i , (19.52) r i = rc n the values are fi = V (ri ), dV r2 dV gi = = c (ri ). dx 2nri dr

(19.53) (19.54)

19.5 Splines for tabulation

541

The values gi can also be determined from a spline ﬁt to fi . In that case the second derivatives are strictly continuous; if the gi ’s are determined from analytical derivatives, the interpolated values approach the analytical function more accurately, but there will be slight discontinuities in the second derivatives at the nodes. For fast recovery from memory fi and gi should be contiguously stored, for example in a two-dimensional array, rather than in separate arrays.8 Recovery from the table We wish to obtain V (r) and F (r) for a given value of r2 , where r2 = (xi − xj )2 + (yi − yj )2 + (zi − zj )2 .

(19.55)

(i) Determine x = nr2 /rc2 , i = int (x) and ξ = x − i. (ii) Fetch fi , gi , fi+1 , gi+1 from the table. (iii) Determine p = 3fi+1 − 3fi − 2gi − gi+1 ,

(19.56)

q = −2fi+1 + 2fi + gi + gi+1 .

(19.57)

(iv) Determine V and dV /dx: V = fi + ξgi + ξ 2 p + ξ 3 q, dV = gi + 2ξp + 3ξ 2 q. dx

(19.58) (19.59)

(v) Reconstruct the force: F =−

2n dV r. rc2 dx

(19.60)

We note that the determination of p and q takes six ﬂoating point operations (ﬂops),9 while the actual interpolation takes ten ﬂops. One may choose to store also pi and qi , thus saving 6 out of 16 ﬂops at the expense of doubling the required memory. The total number of ﬂops is comparable to that required to compute Lennard–Jones potential and forces; for almost any other kind of interaction tables are more eﬃcient than analytical evaluation. Using r2 instead of r causes a rather inappropriate mapping of r onto the table index, the spacing in r being much ﬁner for r close to rc than for values close to the potential minimum, where high accuracy is required. A more 8

9

This requirement depends on the architecture of the computer used; it is certainly true for vector processors that will fetch several words from memory at once. Processors that will hold all tables in cache memory are less demanding in this respect. For example, as follows: u = fi+1 − fi ; q = −2u + gi + gi+1 ; p = −q + u − gi .

542

Splines for everything

x/n 1 0.8 b 0.6 0.4

a

0.2

0.2

0.4

0.6

0.8

1 r/rc

Figure 19.8 Two mappings from distance r to table index x, both avoiding squareroot operations: a shows the quadratic mapping of (19.51), b shows the mapping according to (19.61). For both mappings the intervals in r are indicated when the range of x is divided into 20 equal intervals.

appropriate mapping that still avoids square-root operations, but requires additional ﬂoating point operations, is 2nr2 r2 x= 2 1− 2 . (19.61) rc 2rc Both mappings are represented in Fig. 19.8, showing the better distribution of intervals for the mapping of (19.61) than for the quadratic mapping of (19.51).

19.6 Algorithms for spline interpolation In this section the relevant programs for computing, interpolating and integrating cubic spline functions are given. These programs can be downloaded from the book’s website.10 The general structure is as follows: • Data points are given as list or array data=[x, y], containing two lists or 10

http://www.cambridge.org/9780521835275 or http://www.hjcb.nl.

19.6 Algorithms for spline interpolation

543

arrays x and y of abscissa and ordinates. The x-values must be ordered: xi ≤ xi+1 . For example, a sampled sine function: [[0.,0.5*pi,pi,1.5*pi,2.*pi],[0.,1.,0.,-1.,0.]]. • Data ready for interpolation contain the function values f , as well as the ﬁrst derivatives g: data=[x,f,g]. f is equal to the data ordinates y for splines through the points; for ﬁtting splines yi does not lie on the curve, but fi does. Example (a sampled sine): [[0.,0.5*pi,pi,1.5*pi,2.*pi], [0.,1.,0.,-1.,0.],[1.,0.,-1.,0.,1.]]. • The function cinterpol(x,data) returns the cubic-spline interpolated value f (x) or – if x is an array – an array of interpolated values; the spline is speciﬁed by data=[x,f,g]. • The function cspline(data[,cond]) returns the list [x,f,g] that speciﬁes the spline through data=[x, y]. The optional parameter cond=[[ﬁrst, sec, third ], [ﬁrst, sec, third ]] speciﬁes the extra conditions at the two end nodes: On both the ﬁrst and the last node the values of either the ﬁrst, the second, or the third derivative must be given; the two other values must be given as None. If all values are “None”, a periodic spline is assumed. The default is the natural spline, with: cond=[[None,0.,None],[None,0.,None]]. • The function cintegral(data) returns the values of the integral of the spline function at the nodes. The ﬁrst value is zero; the last value represents the total integral. • The function fitspline(data,weight) produces a natural ﬁtting spline approximating the data points. It uses the argument weight as a general weighting factor for the importance of the minimization of χ2 relative to the minimization of the curvature. A large weight selects a spline through the points, while a small weight tends to perform a least-squares ﬁt to a straight line. The weight is multiplied by the individual weights 1/σi2 ; the standard deviations must be contained in the data. The function returns not only the spline data but also the curvature and the value of χ2 . The iteration to obtain the required deviation is left to the user, who may wish to graphically inspect intermediate results.

python program 19.1 cinterpol(x,data) Cubic interpolation of spline data

544 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35

Splines for everything def cinterpol(x,data): # returns interpolated cubic spline function for argument x # x can b scalar or array # data=[x-values, function values, derivative values] (lists or arrays) xx=array(data[0]) n=alen(xx) if (len(data[1])!=n or len(data[2])!=n): print ’ ERROR: data arrays of unequal length’ print ’ cinterpol aborted’ return None nx=alen(x) def fun(x): i=argmin(abs(x-xx)) if x<xx[i]: i=i-1 x0=xx[i]; x1=xx[i+1] f0=data[1][i]; f1=data[1][i+1] g0=data[2][i]; g1=data[2][i+1] h=x1-x0; g=(f1-f0)/h dx=x-x0 xsqh=dx*dx/h return f0+g0*dx+(3.*g-2.*g0-g1)*xsqh+(-2.*g+g0+g1)*xsqh*dx/h if nx==1: if (x<xx[0] or x>xx[n-1]): print ’ ERROR: x outside range’ print ’ cinterpol aborted’ return None return fun(x) f=empty_like(x) for i in range(nx): z=x[i] if (z<xx[0] or z>xx[n-1]): print ’ ERROR: x[%d] outside range’ % i f[i]=None else: f[i]=fun(z) return f

Comments It is assumed that scipy has been loaded (see page xiii). Lines 06–10 check for equal length of both input arrays. Lines 12–21 deﬁne the interpolated value for a scalar argument; lines 28–34 apply the function for all elements of the input array x.

python program 19.2 cintegral(data) Integral of spline data 01 def cintegral(data): 02 # returns array of integral of cubic spline function at nodes 03 # data=[x-values, function values, derivative values] (lists or arrays) 04 x=data[0]; f=data[1]; g=data[2] 05 n=len(x) 06 if (len(f)!=n or len(g)!=n): 07 print ’ ERROR: data arrays of unequal length’ 08 print ’ cintegral aborted’ 09 return None 10 s=arange(n,dtype=float); s[0]=0. 11 for i in range(n-1): 12 h=x[i+1]-x[i] 13 s[i+1]=s[i]+0.5*(f[i]+f[i+1]+h*(g[i]-g[i+1])/6.) 14 return s

19.6 Algorithms for spline interpolation

python program 19.3 cspline(data[,cond]) Generates ﬁrst derivatives for cubic spline through set of data

01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53

def cspline(data, cond=[[None,0.,None],[None,0.,None]]): # appends derivative array to data=[x,y] using cubic splines through y # x,y can be lists or arrays # returns [x,f,g], f=y (array), g=first derivative array # cond prescribes first, second or third derivatives at end points # default condition is natural splines from linalg import solve x=array(data[0]); f=array(data[1]) n=alen(x)-1 if (alen(f)!=n+1): print ’ ERROR: unequal length of x and y in data’ print ’ cspline aborted’ return None xdif=diff(x) if xdif.min()<=0.: print ’ ERROR: nonincreasing x values’ print ’ cspline aborted’ return None s=1./xdif d=arange(n+1,dtype=float) # predefine d for i in range(1,n): d[i]=2.*(s[i-1]+s[i]) b=arange(n+1,dtype=float) # predefine b for i in range(1,n): b[i]=3.*((f[i]-f[i-1])*s[i-1]**2 + (f[i+1]-f[i])*s[i]**2) if (cond==[[None,None,None],[None,None,None]]): # periodic if (f[0]!=f[-1]): print ’ ERROR: periodic spline with’, print ’ unequal first and last values’ print ’ cspline aborted’ return None else: d[0]=2.*(s[0]+s[n-1]) d=d[:-1] A=diag(d) for i in range(n-1): A[i,i+1]=A[i+1,i]=s[i] A[0,n-1]=A[n-1,0]=s[n-1] b[0]=3.*((f[1]-f[0])*s[0]**2 + (f[n]-f[n-1])*s[n-1]**2) b=b[:-1] g=solve(A,b) g=concatenate((g,[g[0]])) return [x,f,g] head=True # extra first row and column tail=True # extra last row and column if (cond[0][0]!=None): b[1]=b[1]-cond[0][0]*s[0] head=False elif (cond[0][1]!=None): d[0]=2.*s[0] b[0]=3.*(f[1]-f[0])*s[0]**2-0.5*cond[0][1] else: d[0]=s[0] b[0]=2.*(f[1]-f[0])*s[0]**2+cond[0][2]/(6.*s[0]) if (cond[1][0]!=None):

545

546 54 55 56 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72

Splines for everything b[n-1]=b[n-1]-cond[1][0]*s[n-1] tail=False elif (cond[1][1]!=None): d[n]=2.*s[n-1] b[n]=3.*(f[n]-f[n-1])*s[n-1]**2-0.5*cond[1][1] else: d[n]=s[n-1] b[n]=2.*(f[n]-f[n-1])*s[n-1]**2+cond[1][2]/(6.*s[n-1]) if (not head): s=s[1:]; d=d[1:]; b=b[1:] if (not tail): s=s[:-1]; d=d[:-1]; b=b[:-1] A=diag(d) for i in range(alen(d)-1): A[i,i+1]=A[i+1,i]=s[i] g=solve(A,b) if (not head): g=concatenate(([cond[0][0]],g)) if (not tail): g=concatenate((g,[cond[1][0]])) return [x,f,g]

Comments It is assumed that scipy has been loaded (see page xiii). Line 07 imports the scipy package linalg, which contains the linear equation solver solve, used in lines 39 and 69. Although the matrices A are tridiagonal, no use is made of more sophisticated linear solvers for sparse matrices in scipy subpackage linsolve, as the latter are still under development.

python program 19.4 ﬁtspline(data,weight) Generates natural cubic ﬁtting spline, approximating a set of data, given with standard deviations (algorithm adapted from Sp¨ ath, 1973). 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29

def fitspline(data,weight): # computes cubic spline fitting to data # data = [x,y,sig] sig = estimated s.d. of y # x,y,sig can be lists or arrays # weight = overall weight factor attached to data # returns[[x,f,g],[curvature, chisquare deviation]] x=array(data[0]); y=array(data[1]); sig=array(data[2]) n=alen(x)-1 np1=n+1; n1=n-1; n2=n-2; n3=n-3 if (alen(y)!=np1 or alen(sig)!=np1): print ’ ERROR: unequal lengths in data’ print ’ FitSpline aborted’ return None q=arange(np1,dtype=float); p=arange(np1,dtype=float) g=arange(np1,dtype=float) y2=arange(np1,dtype=float); y2[0]=y2[n]=0. w=weight/sig**2 f=concatenate((diff(x),[0.])) q1=q2=q[0]=p[0]=p[1]=0. j1=0 w1=1./w[0]; w2=1./w[1] h=0.; h1=f[0]; h2=f[1] r1=(y[1]-y[0])*h1 for k in range(n1): k1=k+1; k2=k+2 if (k>0): h=p1-q1*q[j1] h3=f[k2] w3=1./w[k2] s=h1+h2

19.6 Algorithms for spline interpolation 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76

547

t=2./h1+2./h2+6.*(h1*h1*w1+s*s*w2+h2*h2*w3) r2=(y[k2]-y[k1])*h2 p2=1./h2-6.*h2*(w2*s+w3*(h2+h3)) q3=6.*w3*h2*h3 z=1./(t-q1*p[k]-h*q[k]) if (k0): r=r-h*g[j1] if (k>1): r=r-q1*g[j2] g[k]=z*r j2=j1; j1=k q1=q2; q2=q3 p1=p2 h1=h2; h2=h3 w1=w2; w2=w3 r1=r2 y2[n1]=g[n2] if (n2!=0): y2[n2]=g[n3]-q[n2]*y2[n1] if (n3!=0): for j in range(n3): k=n3-j y2[k]=g[k-1]-q[k]*y2[k+1]-p[k+1]*y2[k+2] h1=0 for k in range(n): j2=k+1 g[k]=f[k] h2=f[k]*(y2[j2]-y2[k]) q[k]=h2/6. f[k]=y[k]-(h2-h1)/w[k] p[k]=0.5*y2[k] h1=h2 f[n]=y[n]+h1/w[n] for k in range(n): j2=k+1 h=g[k] g[k]=(f[j2]-f[k])*h-(y2[j2]+2.*y2[k])/(6.*h) g[n]=(f[n]-f[n1])*h+(2.*y2[n]+y2[n1])/(6.*h) curv=0. # compute curvature for k in range(n1): curv=curv+q[k]*(f[k+1]-f[k]) curv=-6.*curv print ’curvature = ’, curv, ’ ’, chisq=0. # compute chisquare sum for k in range(np1): chisq=chisq+((y[k]-f[k])/sig[k])**2 print ’chisquare = ’, chisq return [[x,f,g],[curv,chisq]]

Comments This program is adapted from Sp¨ ath (1973) and contains an explicit linear solver; it does not require the package linalg. A trial weight must be given; the program returns both the curvature and the chi-square sum. The weight should be adjusted such that the chi-square sum is statistically acceptable, i.e., near the number of data points.

548

Splines for everything

19.7 B-splines functions11

(p) Ni (t)

are local smooth functions of a parameter t (p B-spline is the order of the spline), which are used to construct smooth B-spline curves in one- or multidimensional space by linear superposition, given a set of control points (also called de Boor points) di at the nodes ti : (p) F (t) = Ni (t)di . (19.62) i

Each B-spline function is a polynomial with local support, i.e., it has (positive) values only on a limited local interval and is zero outside this interval. The support depends on the order p of the spline and is equal to the interval [ti , ti+p ]. The higher the order p, the smoother the curve, but the larger its support. If the control points are vectors in 3D space, a curve in 3D space results; the control points may also be real or complex scalars that reconstruct a (real or complex) function f (x). B-spline surfaces result from a product of spline functions with two parameters u and v and a two-dimensional set of control points: (p) (p) F (u, v) = Ni (u)Nj (v)dij . (19.63) i,j

The B-spline functions are normalized if they integrate to unity. (p) The normalized B-spline function Ni is deﬁned recursively: (1)

(19.64) p = 1: Ni (t) = 1 for ti ≤ t < ti+1 ; 0 elsewhere, ti+p − t t − ti (p) (p−1) (p−1) p ≥ 2: Ni (t) = N (t) + N (t).(19.65) ti+p−1 − ti i ti+p − ti+1 i+1 Most common are the cardinal B-splines, which have equidistant intervals ti+1 − ti , and especially with distance 1 (“normalized nodes”). The nodes (or knots) are then a set of consecutive integers. They produce uniform B-spline curves. These cardinal B-spline functions are thus deﬁned as (1)

p = 1: Ni (t) = 1 for i ≤ t < i + 1; 0 elsewhere, 11

(19.66)

See Engeln-M¨ ullges and Uhlig (1996). B-splines are also introduced in Essmann et al. (1995). The name derives from B for “basis”. Although in principle already known by Laplace and Euler, B-spline functions and their applications in approximation theory were developed in the 1960s and 1970s. The basic mathematical theory can be found in Schumaker (1981), but a more practical guide is a published lecture series by Schoenberg (1973). A book by de Boor (1978) provides Fortran programs. A relatively more modern description can be found in Chapter 4 of Chui (1992). There is no universally agreed notation for B-splines functions. Our notation N is most common for the normalized cardinal spline functions, although the meaning of suband superscripts may diﬀer. Schoenberg (1973) denotes non-normalized B-spline functions by Q and central B-spline functions, which are shifted to have the origin at the maximum of the function, by M . The terms knots and nodes are both used for the control points. One should distinguish between B-splines (curves or surfaces) and B-spline functions. The former are linear combinations of the latter and could be called B-spline series.

19.7 B-splines (p)

p ≥ 2: Ni (t) =

549

t − i (p−1) i + p − t (p−1) N Ni+1 (t). (t) + p−1 i p−1

(19.67)

Clearly Ni is simply translated over i units with respect to N0 : (p)

(p)

(p)

Ni (t) = N0 (t − i); support of Ni (t) : i ≤ t < i + p.

(19.68)

We may omit the subscript 0 and thus deﬁne the basic B-spline functions (p) N (p) (t) for the support 0 ≤ t < p, from which all other functions Ni (t) are derived by translation, as p = 1: N (1) (t) = 1 for 0 ≤ t < 1; 0 elsewhere, t p − t (p−1) N (p−1) (t) + N (t − 1). p ≥ 2: N (p) (t) = p−1 p−1

(19.69) (19.70)

The functions are symmetric around t = p/2, where they attain a maximum. Some texts deﬁne centralized B-spline functions by taking the origin at p/2; for odd p this has the disadvantage that the knots do not occur at integer arguments. The B-spline functions deﬁned here are also called forward Bspline functions (Silliman, 1974). Figure 19.9 shows the ﬁrst ﬁve normalized cardinal splines for i = 0. The functions N (p) (t) do not only integrate to 1, but the sum of their values at the nodes equals unity as well: i+p

(p)

Ni (j) = 1.

(19.71)

j=i

Thus they, so to say, distribute the number 1 among points close to the center p/2. An interesting property of cardinal B-spline functions is that they represent the distribution generated by the sum of p independent random numbers η, each homogeneously distributed in the interval 0 ≤ η < 1. As p becomes larger, according to the central limit theorem, this distribution tends to a normal distribution with mean p/2 and variance p/12. Therefore

(t − p/2)2 1 p (p) lim N (t) = √ exp − , σ2 = . (19.72) 2 p→∞ 2σ 12 σ 2π This property is a direct consequence of a general convolution theorem for B-spline functions: each N (p) is the convolution of N (p−q) (1 ≤ q < p) with N (q) . For the special case q = 1 N (q) is the unit pulse function and the convolution becomes another recurrent relation 1 t (p) (p−1) (1) (p−1) N (t) = (N ∗ N )(t) = N (t − t ) dt = N (p−1) (t ) dt . 0

t−1

(19.73)

550

Splines for everything 1

N0(1)

p=1 1

N0(2)

p=2

1/2

1/2

N0(3)

p=3

2/3 1/6

p=4

1/6

11/24

−1

11/24

1/24

p=5

0

1

N0(4)

N0(5)

1/24

2

3

4

5

6

t (p)

Figure 19.9 The ﬁrst ﬁve normalized cardinal B-spline functions N0 (t).

The convolution theorem implies that the Fourier transform of N (p) is the product of the Fourier transforms of N (q) and N (p−q) . Since the Fourier transform of the unit pulse N (1) is proportional to sin 12 ω/ω (ω is the conjugate variable in reciprocal space),12 the Fourier transform of N (p) is proportional to the p-th power of that function, disregarding a phase factor due to the shifted origin. This transform decays faster in reciprocal space for higher-order splines. A B-spline function of order p consists of piecewise polynomials of degree p − 1 and is p − 2 times continuously diﬀerentiable; thus N (4) (t) is twice diﬀerentiable, which makes it (and higher-order functions) suitable to express energy functions: the ﬁrst derivative (needed for forces) is suﬃciently smooth as it has a continuous derivative. For the derivatives the following 12

This applies to the central function; the Fourier transform of the function N (1) contains a phase factor exp( 12 iω) because the origin is shifted with respect to the symmetry axis.

19.7 B-splines

551

relation holds: (p)

dN0 (t) (p−1) (p−1) = N0 (t) − N1 (t). dt

(19.74)

B-spline curves are not interpolating, as the control points are not equal to the nodal values. Each nodal value is composed of nearby control points according to (19.62). One may construct interpolating curves by deriving the control points from a corresponding set of equations. For closed, i.e., periodic, B-spline curves, function values are periodic on [0, n) and there are exactly n nodes. One therefore needs n known values to solve for the coeﬃcients ci . When the known values are the nodal function values, an interpolating periodic spline is obtained: f (t) =

n−1 i=0

ci

(p)

Ni+kn (t).

(19.75)

k∈Z

The sum over k takes care of the periodicity, but only a few values are included because of the compact support of the functions. For example, the interpolating fourth-order B-spline for a sine function sin 12 πt on the periodic interval [0, 4) (Fig. 19.10), given the nodal values (0, 1, 0, −1) at the nodes (0,1,2,3) is 3 3 f (t) = (N−1 + N3 ) − (N1 + N5 ). (19.76) 2 2 These coeﬃcients are the solution of the following ⎛ ⎞ ⎛ ⎞⎛ 0 1 4 1 c0 ⎟ ⎜ ⎟⎜ 1⎜ ⎜ 1 0 1 4 ⎟ ⎜ c1 ⎟ = ⎜ ⎝ ⎠ ⎝ c2 ⎠ ⎝ 4 1 0 1 6 c3 1 4 1 0

set of equations: ⎞ 0 1 ⎟ ⎟, (19.77) 0 ⎠ −1

and have the values (0, −3/2, 0, 3/2). The supporting B-spline functions are depicted in Fig. 19.10. Not surprising this B-spline is identical to the cubic interpolating spline shown in Fig. 19.2 on page 531. The recurrence relation (19.67) allows the following simple algorithm to compute spline values (Griebel et al., 2003). The value of p must be an integer not smaller than 2. While a recurrent program is elegant, a nonrecurrent version would be more eﬃcient. python program 19.5 Bspline Computes the value of the cardinal B-spline function N (p) (t) of real t and integer order p ≤ 2.

552

Splines for everything

sin (πt/2) N−3(t) c1 = −3/2 N−2(t) c2 = 0 N−1(t) c3 = 3/2

−2

0

2

4

6

N0(t)

c0 = 0

N1(t)

c1 = −3/2

N2(t)

c2 = 0

N3(t)

c3 = 3/2

t

Figure 19.10 The sine function, periodic on the interval [0, 4), approximated by a fourth-order B-spline.

01 def Bspline(t,p): 02 if (t<=0. or t>=p): return 0. 03 if (p==2): return 1.-abs(t-1.) 04 return (t*Bspline(t,p-1) + (p-t)*Bspline(t-1.,p-1))/(p-1.) Comments Recurrence is pursued until the order p = 2 is reached, which is the “hat-function” speciﬁed in line 2.

For non-periodic (open) B-splines that are to be ﬁtted on a function known at n + 1 points t = 0, . . . , n, one needs coeﬃcients for all the B-spline functions that share their support with the interval [0, n), which are n + p − 1 in number. This means that, in addition to the n + 1 nodal values, (p − 2) extra conditions must be speciﬁed. These can be points outside the interval or ﬁrst or second derivative speciﬁcations. Instead of function values at the nodes, values at arbitrary (e.g., random) arguments may be speciﬁed, as long as they provide a suﬃcient number of independent data to solve for the coeﬃcients. While in general a set of equations must be solved to ﬁnd the spline

19.7 B-splines

553

coeﬃcients, there is an exception for exponential functions. Functions of the type f (t) = z t , among which the important class of exponentials f (t) = exp(iωt), can be approximated by a B-spline series ∞

f (t) ≈

cn Nn(p) (t).

(19.78)

n=−∞

These are called Euler exponential splines (Schoenberg, 1973; Silliman, 1974). We wish to ﬁnd the coeﬃcients cn such that the spline series equals f (t) = z t exactly at all integer values of t. Now consider f (t + 1). On the one hand we may expand the shifted series as follows: f (t + 1) ≈

∞

cn Nn(p) (t

+ 1) =

n=−∞

∞

(p) cn Nn−1 (t)

n=−∞

=

∞

cn+1 Nn(p) (t),

n=−∞

(19.79) but, on the other hand: f (t + 1) = zf (t).

(19.80)

Therefore there is a relation between coeﬃcients: cn+1 = zcn .

(19.81)

Denoting c0 by c, it follows that cn = z n c, and f (t) ≈ c

∞

z n Nn(p) (t).

(19.82)

(19.83)

n=−∞

The inﬁnite sum takes care of the periodicity. The constant c follows from the requirement that the approximation is exact at the nodes t = k, k ∈ Z: ∞

zk = c

z n Nn(p) (k).

(19.84)

n=−∞

This can be transformed by replacing n − k by n and shifting to the spline function N = N0 , yielding −1

p−1 −n (p) z N (n) . (19.85) c= n=1

Thus in (19.83) and (19.85) we have obtained the solution for an interpolating exponential spline. These exponential splines are particularly useful to approximate terms in a

554

Splines for everything imag 1

t 012

K

(a)

t

2

0

real

p=4 exp(0.5πit) (b)

3

Figure 19.11 (a) the t-axis discretized in K = 16 intervals of unit length for use in FFT. The function to be transformed is periodic on [0, K). The supports of the four fourth-order interpolating spline functions, needed to approximate exp(2πimt/K), is indicated by four lines. (b) Fourth-order spline interpolation (drawn curve) for exp(0.5πit) (dashed circle).

Fourier sum, such as exp(iωt) (for arbitrary t), when fast Fourier transforms (FFT) are used, deﬁned only at discrete values of ω and t. Take for example a periodic t-axis divided into K (K is an even integer) intervals of unit length. In FFT (see page 324) the angular frequencies are discretized in K steps K K K 2πm ωm = ; m = − , − + 1, . . . , − 1, (19.86) K 2 2 2 and therefore one needs to evaluate exp(2πimt/K) for |m| ≤ K/2. Only the values t = n, n = 0, . . . K − 1 can be handled by the FFT, and thus one wishes to express exp(2πimt/K) in an interpolating spline series with known values at integer values of t. This is accomplished by (19.83) and (19.85): ∞ 2πimn 2πim t ≈ c Nn(p) (t), exp (19.87) exp K K n=−∞ −1

p−1 2πimn N (p) (n) exp − . (19.88) c = K n=1

Figure 19.11(a) shows the t axis for the example K = 16 and the supports for the four 4-th order spline functions needed for the indicated value of t. The quantity to be Fourier transformed is thus distributed over the four preceding integer values. In Fig. 19.11(b) the fourth-order spline approximation (drawn curve) of the function exp(0.5πit) (providing four points per period)

19.7 B-splines

555

is compared with the exact function (dashed circle) to show the accuracy of the interpolation (better than 0.03). The absolute error in the approximation equals approximately (|ω|/π)p (Schoenberg, 1973; Silliman, 1974). This makes the approximation quite imprecise for the highest frequency ωmax = π (m = K/2) that occurs in the FFT. In fact, for the highest frequency the exponential exp(iπn) is alternating between +1 and −1, and the resulting spline is fully real. A fourth-order spline does reproduce the real part cos πt of exp(πt) fairly well (within 0.02) but fails completely to reproduce the imaginary part. For odd values of the spline order p there is no solution since the sum evaluates to 0/0 (e.g., for p = 5 the sum would be 2/24 − 22/24 + 22/24 − 2/24 = 0). This combination of values must be excluded. In usual applications the grid is chosen suﬃciently ﬁne for the highest frequency terms to be of vanishing importance.

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Index

a.u., see atomic units ab initio molecular dynamics, 151 Abelian group, 380 absolute activity, 462 acid dissociation constant, 439 action, in mechanics, 398 activity coeﬃcient, 435 activity, in thermodynamic potential, 436 adiabatic limit, 132 advancement, reaction, 448 aﬃnity, 448 AFM, see atomic force microscope alchemy, computational, 225 alias, 324 AM1, see Austin model 1 Ampere’s law, 337 Andersen thermostat, 198 Angstrom (unit), xv angular acceleration, 410 angular momentum deﬁnition, 405 equation of motion, 405 angular velocity, 409 annealing, 205 associated Legendre functions, 361 atmosphere (unit), xv atomic force microscopy, 237 atomic units, xvii, xvii au, see atomic units Austin model 1, 106 autocorrelation function, 317, 325 axial vectors, 409 Axilrod–Teller eﬀect, 187 azurin, 329 B-splines, 548 back reaction mean ﬁeld, 131 surface hopping, 134 Banach space, 381 bar (unit), xv BE, see Bose-Einstein beables, 64

Beeman algorithm, 192 beta identiﬁcation with temperature, 458 binary rotations, 414 Bohmian dynamics, 64 Bohr magneton, 34 Bohr, Copenhagen interpretation, 65 boiling point elevation, 442 Boltzmann statistics, 470 Born energy charge in cavity, 343 in dielectric medium, 344 with ionic strength, 345 Born reaction potential, 345 Born–Oppenheimer approximation, 97 Bose–Einstein statistics, 467 boson, 37 boundary conditions, dielectric, 342 boundary conditions artifacts PBC, 145 conducting, 166 continuum, 148 periodic, 141 restrained shell, 149 tin-foil, 166 triclinic unit cell, 143 boundary element method, 342 Brownian dynamics with acceptance criterion, 273 Brownian dynamics, 268 Bspline, Python program, 551 Caley–Hamilton relation, 389 Caley–Klein parameters, 407 candela (unit), xxiv canonical distribution cartesian coordinates, 497 generalized coordinates, 498 quantum, 457 canonical partition function, 457 cap atom, 189 Car–Parrinello method, 151 cardinal B-splines, 548

587

588 CASSCF, see complete active space SCF center of mass, 404 CGPM, xv charge spread function cubic, 369 Gaussian, 368 general, 363 chemical potential, see thermodynamic potential chi-square distribution, 532 chi-square table, 532 cintegral, Python program, 544 cinterpol, Python program, 543 Clausius’ virial, 485 Clausius–Clapeyron equation, 435 coalescing drops, DPD, 309 coarse graining, 249 coarse-graining weight function, 289 coexistence region, 434 colligative properties, 441 commutator, 384 complete active space SCF, 105, 136 compressibility adiabatic, 286, 426 in ﬂuid dynamics, 286 in phase space, 496 isothermal, 308, 426 phase space, 198 computational alchemy, 224 conducting boundary conditions, 166 conformation, 204 conical intersection, 136 constraint matrix, 421 constraints coordinate resetting, 419 ﬂexible, 158 force, 418 generalized coordinates, 418 holonomic, 417 hypersurface, 418 in MD algorithm, 194 in thermodynamic integration, 226 Lagrange multiplier, 418 LINCS, 422 linear, 422 metric tensor eﬀect, 499 projection method, 418, 421 rotational, 145 SHAKE, 420 continuity equation, 282 continuum corrections, 168 convolution, 316 coordinate four-vector, 26 coordinate transformations, 384 Copenhagen interpretation, 65 corrections continuum environment, 168 correlation function, classical, 511 Coulomb energy, 340 coupling parameter, 221 Crank–Nicholson algorithm, 116

Index cspline, Python program, 545 cubic splines, 526 cyclic coordinates, 400 DAC, see divide-and-conquer, de Broglie wavelength, 469 Debye (unit), xv Debye length, 342 Debye–H¨ uckel approximation, 341 degree of advancement, 448 delta-response, 507 density functional theory description, 102 time-dependent, 105 density matrix and partition function, 394 coordinate representation, 394 deﬁnition, 392 ensemble-averaged, 394 time evolution, 392 transformation, 393 detailed balance in Monte Carlo, 13, 273 in replica exchange, 206 in smart Monte Carlo, 273 DFT, see density functional theory dielectric boundary conditions, 342 dielectric displacement, 336 dielectric properties, from ﬂuctuations, 514 diﬀerential relations, thermodynamics, 431 diﬀerential exact, 426 total, 426 diﬀusion constant from Einstein relation, 519 from NEMD, 520 diﬀusive systems, 214 dimerization constant, 438 dipole correction, lattice sum, 372 Dirac equation, 27 displacement current density, 337 dissipative particle dynamics, 305 divide-and-conquer, 106 double well, 82 DPD, see dissipative particle dynamics dummy particles, 157 dynamics Brownian, 268 dissipative particle, 305 ﬂuid continuum, 281 Langevin, 252 mesoscopic, 249, 297 molecular, 139 non-inertial, 268 replica-exchange, 205 scaled particle, 306 dyne (unit), xv eﬀective Born radius, 353 eﬀective pair potentials, 174 Ehrenfest

Index classical limit, 43 Einstein relation, 519 electric constant, xvi electric dipole density, 336 electric factor, xviii electric ﬁeld deﬁnition, 335 fast multipole methods, 362 hierarchical methods, 362 in conducting medium, 348 long-range contribution, 365 long-range Fourier solution, 366 of charge in cavity, 343 periodic systems, 362 short-range contribution, 364 electron acceptor, 439 electron donor, 439 electron transfer reaction, 439 electrostatic unit, 377 electrostatics, 340 energy density, electromagnetic, 339 ensemble canonical, 460 grand-canonical, 461 isobaric-isothermal, 460 microcanonical, 460 size eﬀects, 462 enthalpy, deﬁnition, 426 entropy production in reaction, 448 irreversible, 444 entropy ﬁnite-size eﬀects, 463 in statistical mechanics, 458 thermodynamic deﬁnition, 426 EOS, see equation of state equation of state, 433 equilibrium constant, 438 equipartition theorem, 502 equipartition generalized, 503 kinetic energy, 503 erg (unit), xv ergodic postulate, 454 essential dynamics, 216 esu, see electrostatic unit Euler angles, 412 Euler equations, 407 Euler exponential splines, 553 Euler parameters, 413 Euler’s theorem, 414 Euler–Rodrigues parameters, 413 Eulerian derivative, 281 Ewald summation, 368 exact diﬀerential, 426 excess pressure, coarse-graining, 294 excess thermodynamic potential, 219 exchange potential, 479 exchange-correlation potential, 104 excited-state quantum mechanics, 105 expectation, observable, 391

exponential matrices, 385 exponential operators, 385 exponential splines, 553 Faraday (unit), 440 Faraday’s induction law, 337 fast Fourier transform, 323 fast multipole method, 168 FD, see Fermi-Dirac FENE model, 256 Fermi–Dirac statistics, 467 fermion, 37 Feynman path integral, 44 Feynman–Hibbs potential, 71 FFT, see fast Fourier transform ﬁne-structure constant, xviii ﬁtspline, Python program, 546 ﬁtting splines, 530 Fixman’s theorem, 501 ﬂexible constraint, 158 ﬂuctuation–dissipation theorem ﬁrst, 259, 511 in Langevin dynamics, 257 restricted form, 258 second, 260 ﬂuid dynamics coarse-graining, 288 continuity equation, 282 energy balance, 288 equation of motion, 282 equation of state, 285 equations, 281 Eulerian derivative, 281 heat conduction, 287 incompressible ﬂuids, 285 Lagrangian derivative, 281 momentum conservation, 283 momentum ﬂux, 283 stress tensor, 283 viscosity, 285 viscous stress tensor, 284 FMM, see fast multipole method Fock operator, 101 Fokker–Planck equation, 268 for Brownian dynamics, 272 for diagonal friction, 272 for generalized Langevin, 271 for velocity, 268 general equation, 270 force ﬁeld bond angle, 154 bonded interactions, 153 Buckingham potential, 156 constraints, 155 Coulomb interaction, 156 dihedrals, 154 eﬀective potentials, 174 empirical adjustments, 150 ﬂexible constraint, 158 ideal, 183

589

590 improper dihedral, 155 Lennard–Jones, 155 long-range Coulomb, 164 long-range dispersion, 159 Morse potential, 154 multipoles, 158 parent model, 187 polarizability, 157 polarizable, 171 pressure correction, 161 QM/MM, 188 quartic bond potential, 154 reaction ﬁeld, 164 Ryckaert–Bellemans potential, 155 shifted potential, 160 switching function, 160 truncated potential, 160 united atom, 153 force ﬁelds, 149 four-dimensional dynamics, 225 four-vectors, 26 Fourier transform convolution, 316 deﬁnitions, 315 discrete, 323 FFT, 323 Gaussian function, 322 multidimensional, 331 square pulse, 320 triangular pulse, 321 free energy, see also potential of mean force free energy by integration, 222 by perturbation, 222 by slow growth, 223 by spatial integration, 213 non-diﬀusive systems, 213 quantum correction, 477 freezing point depression, 442 frequency response, 508 friction kernel, 252 friction tensor, 274 friction and noise, 259 from diﬀusion, 275 from Stokes’ law, 275 hydrodynamic interaction, 276 solute particle, 275 frictional force in Langevin dynamics, 251 FT, see Fourier transform fugacity, 435 gauge invariance, 338 Gauss equation, 337 Gauss thermostat, 198 GB, see generalized Born solvation model Gear algorithm, 191 generalized Born solvation model, 352 generalized equipartition theorem, 503 generalized Langevin equation, 253

Index generalized momentum, 399 GENERIC formalism, 306 ghost particle, 219 Gibbs free energy, deﬁnition, 426 Gibbs function, see Gibbs free energy Gibbs’ paradox, 453, 470 Gibbs’ phase rule, 433 Gibbs–Duhem relation, 429, 447 Gibbs–Helmholtz relation, 430 grand ensemble, 461 grand-canonical ensemble, 460 Green’s function QMC, 96 Green’s function, 45 GROMACS, xii group velocity, 24 half-reaction, electrochemical, 439 haloalkane dehydrogenase, 172 Hamiltonian classical, 399 dipole in ﬁeld, 35 generalized, 402 quantum, 43 relativistic, 28 with vector potential, 32 Zeeman, 34 hartree (unit), xviii Hartree-Fock equation, 101 limit, 101 methods, 99 heat capacity isobaric, 426 isochoric, 426 heat conduction coeﬃcient, 287 Heisenberg interpretation QM, 65 uncertainty relation, 22, 319 helium quantum eﬀects, 41 Quantum Monte Carlo, 92 Hellmann–Feynman force, 131 Helmholtz free energy, deﬁnition, 426 Helmholtz free energy, 459 Helmholtz function, see Helmholtz free energy Henry’s constant, 438 hermitian conjugate, 382 hermitian operator, 383 hidden variables, 65 Hilbert space, 381 Hohenberg–Kohn ﬁrst theorem, 102 second theorem, 103 hydrodynamic interaction, 276 hydrodynamics classical, 281 quantum, 64 hydrogen ﬂuoride classical vibration, 6 quantum vibration, 87 hydrogen

Index quantum eﬀects, 41 hypercomplex numbers, 413 hyperpolarizability, 172 ideal solution, 436 image charge in sphere, 350 imaginary time path integral, 51 inch (unit), xv inertia tensor, 408 information and Boltzmann H-function, 455 in statistical mechanics, 455 Shannon, 455 integrating factor, 426 ion hydration correction, 170 ionic strength, 342 irrelevant degrees of freedom, 249 irreversible processes entropy production, 444 Onsager relations, 450 reactions, 448 stationary state, 450 thermodynamic forces, 444 thermodynamics, 444 isobaric-isothermal ensemble, 460 isokinetic thermostat, 198 Itˆ o-Stratonovich dilemma, 254 Itˆ o, 254 Jacobian, 495 Jarzynski equation, generalized, 241 Jarzynski equation, 239 Jaynes, information, 455 Joule heat, 339 Joule–Thomson coeﬃcient, 426 kcal (unit), xv Kelvin (unit), xxiv kilogram (unit), xxiv kilogramforce (unit), xvii Kirkwood quantum correction, 476 Kohn-Sham orbitals, 103 Kramers equation, 271 Kramers–Kronig relations, 509 Lagrange equations, 398 Lagrange multipliers, 456 Lagrangian derivative, 281 Lagrangian, 398 Lande g-factor, 34 Langevin equation generalized, 253 Markovian, 261 pure, 258 simple pure, 267 simple, 254, 267 Laplace equation, 346 leap-frog algorithm, 192 Legendre polynomials, 360 length, SI unit of, xxiv

591

Lie–Trotter–Suzuki expansion, 386 light intensity, SI unit of, xxiv linear constraint solver, 422 linear response, 506 linear scaling methods, 106 Liouville equation classical, 492 generalized, 494 proper, 494 Liouville operator classical, 493 generalized, 493 Liouville–von Neumann equation, 392 Lorentz convention, 338, 33 Lorentz force, 335 Lorentz transformation, 25 Lyapunov exponent, 204 Madelung ﬂuid, 64 magnetic dipole density, 336 magnetic ﬁeld deﬁnition, 335 magnetic intensity, 336 many-electron quantum methods, 98 Markovian Langevin equation, 261 mass ﬂux density, deﬁnition, 282 mass tensor, 401, 498 mass, SI unit of, xxiv mass-metric tensor, 401 material derivative, 281 maximum entropy, 215 Maxwell relations, 431 Maxwell’s equations integrated form, 337 polarizable media, 336 vacuum, 335 MC, see Monte Carlo simulations MCDHO model, see water MD, see molecular dynamics mdstep, Python program, 8 mean-ﬁeld back reaction, 131 mercury relativistic eﬀects, 30 mesh Ewald methods, 168 mesoscopic dynamics, 249, 297 meter (unit), xxiv metric tensor eﬀect of constraints, 502 metric tensor, 143 Metropolis Monte Carlo, 272 microcanonical ensemble, 459 Minkovsky space, 25 MNDO, see modiﬁed neglect of diﬀerential overlap mobility matrix, 276 modiﬁed neglect of diﬀerential overlap, 106 mol (unit), xxiv mole, xxiv molecular dynamics, 139 molecular units, xviii, xvii moment of inertia, 408

592 momentum density, 283 momentum ﬂux tensor, 481 momentum four vector, 26 Monte Carlo simulations detailed balance, 13 diﬀusional quantum, 89 Green’s function quantum, 96 Metropolis scheme, 272 multicanonical, 205 path integral, 61 references, 13 replica exchange, 205 smart MC, 273 variational quantum, 84 MOPAC, 106 Morse oscillator classical, 6 quantum, 87 Mulliken analysis, 156 multi-level system, 128 multicanonical Monte Carlo, 205 multiple time-step algorithms, 193 multipole expansion, 353 multipoles deﬁnition, 356 in force ﬁeld, 158 in spherical harmonics, 359 traceless, 358 Navier–Stokes equation general, 285 incompressible ﬂuids, 285 NDDO, see neglect of diatomic diﬀerential overlap neglect of diatomic diﬀerential overlap, 106 NEMD, see non-equilibrium molecular dynamics non-diﬀusive systems, 214 non-equilibrium molecular dynamics, 519 non-inertial dynamics, 268 Nos´ e–Hoover thermostat, 201 nuclear quantum states, 107 Numerov’s method, 80 numerov, Python program, 80 Nyquist’s theorem, 324 octupole moment, 356 octupole, traceless, 359 Onsager, reciprocal relations, 450 open systems, 433 operator commuting, 384 eigenvalue equation, 383 energy, 22 exponential, 385 hermitian, 383 in Fourier space, 317 in Hilbert space, 382 momentum, 22 projection, 388 quantum mechanical, 22

Index wave vector, 317 Oseen tensor, 276 osmolality, 441 osmotic coeﬃcient, 437 oversampling, 324 oxidant, 439 parameterized model 3, 106 parity, 36 Parseval’s theorem, 317 partial diﬀerential relations, thermodynamics, 431 partial molar quantities, 427 particle insertion, 219, 237 particle mesh Ewald method, 168, 374 particle–particle particle–mesh, 374 partition function classical, 476 quantum correction, 476 quantum, 457 trace of matrix, 465 path integral classical limit, 48 concept, 44 equivalence Schr¨ odinger eqn, 47 free particle, 55 harmonic potential, 57 imaginary time, 51 minimum action, 48 molecular dynamics, 61 Monte Carlo, 61 normalization, 49 string of beads, 62 Pauli exclusion principle, 37 parity rule, 37 PBC, see periodic boundary conditions PCA, see principal component analysis periodic boundary conditions, 141 periodic cubic splines, 526 periodic disturbance, 508 perturbation time-dependent, 120 time-independent, 119 pH, deﬁnition, 439 phase diagram, 434 phase rule, 433 phase space compressibility, 496 deﬁnition, 492 incompressible, 494 metric tensor, 495 phase velocity, 20 phenomenological coeﬃcients, 449 phenomenological relations, 449 photoactive yellow protein, 136 pK, deﬁnition, 439 PM3, see parameterized model 3 PME, see particle mesh Ewald method PMF, see potential of mean force Poisson bracket, 496

Index Poisson equation, 341 Poisson–Boltzmann equation full, 341 linearized, 342 polarizability anisotropic, 176 choices, 176 energies and forces, 181 ﬂuctuating charges, 176 in force ﬁeld, 157 induced dipoles, 176 shell model, 176 Thole model, 177 units, xvi positrons, 28 potential of mean force and free energy, 227, 231 and metric, 231 deﬁnition, 230 derivation, 233 harmonic well, 231 in Langevin equations, 255 nonequilibrium, 239 particle insertion, 237 quantum corrections, 232 thermodynamic integration, 234 thermodynamic perturbation, 235 umbrella sampling, 235 potential-derived charges, 156 potential scalar, 337 vector, 337 poundforce (unit), xvii Poynting vector, 338 PPPM, see particle-particle particle-mesh pressure and virial, 485 average, 484 diatomic example, 491 from c.o.m., 489 internal, 432 kinetic, 432 localization, 483 relation to stress, 481 tensor, 481 thermodynamic, 485 principal component analysis, 216 proton acceptor, 439 proton donor, 438 proton transfer reaction, 438 pseudovectors, 409 psi, Python program, 81 psips, 89 Pulay force, 131, 133 pulling simulations, 237 pure Langevin equation, 258 Python, essentials, xii Python program Bspline, 551 cintegral, 544 cinterpol, 543

593 cspline, 545 ﬁtspline, 546 mdstep, 8 numerov, 80 psi, 81 spectrum from time series, 329 SRstep, 87 walkers, 94

QM/MM, 188 QMC, see quantum Monte Carlo quadrupole moment, 356 quadrupole, traceless, 358 quantity, SI unit of, xxiv quantum chemistry, 98 quantum correction, as potential, 478 quantum correction exchange, 479 free energy, 477 harmonic oscillator, 58, 74 partition function, 476 quantum dynamics, back reaction, 109 quantum dynamics adiabatic, 110 crossing states, 111 grid integration, 115 mean ﬁeld, 112 non-stationary potential, 114 solvated electron, 118 surface hopping, 113 quantum force, 65 quantum hydrodynamics, 64 quantum Monte Carlo diﬀusional, 89 Green’s function, 96 helium, 92 variational, 84 quantum potential, 68 quantum stochastic dynamics, 64 quantum stress tensor, 69 quantum width, 40 quaternions and rotation matrices, 416 deﬁnition, 413 equation of motion, 416 normalized, 413 random force in Brownian dynamics, 269 in Langevin dynamics, 251 random phase approximation, 394 Raoult’s law, 442 reaction coordinate, 230 reaction equilibria, 437 reaction ﬁeld gradient, 349 reaction ﬁeld dipole in cavity, 347 from charges in cavity, 348 from excentric charge in cavity, 349 gradient from quadrupole, 349 in dipole ﬂuctuation, 515

594 in MD, 164, 165 with ionic strength, 347 reaction ﬂux, 448 reaction potential from charge in cavity, 344 from charges in cavity, 348 from excentric charge in cavity, 349 from image charge, 350 in dielectric medium, 344 with ionic strength, 345 reciprocal lattice vectors, 332 reciprocal lattice, Fourier space, 332 reciprocal relations, 450 Redﬁeld limit, 125 reductant, 439 relative dielectric permittivity, 337 relativistic correction, 29 relativistic quantum mechanics, 25 relaxation methods quantum, 85 relevant degrees of freedom, 249 REM, see replica exchange method REMD, see replica exchange molecular dynamics replica exchange method, 205, 211 replica exchange molecular dynamics, 205 replica Monte Carlo, 205 RESPA, 191 reversible processes, 425 rhombic dodecahedron, 142 rigid body motion, 405 Rodrigues parameters, 413 rotation matrix and quaternions, 416 deﬁnition, 406 time derivative, 409 rotation Euler angles, 412 quaternions, 413 unit vectors, 411 rotational constraint, 145 rotational partition function, 471 rotation, 404 rovibrational states, 107 saturation pressure, 435 scaled particle hydrodynamics, 306 SCF, see self-consistent ﬁeld Schr¨ odinger equation time-independent, 78 imaginary time, 86 time-dependent, 23 Schwarz inequality, 319 second (unit), xxiv second virial coeﬃcient deﬁnition, 436 quantum corrections, 72 quantum statistics, 469 water, 175 self-consistent ﬁeld, 101 self-energy, 340

Index semi-empirical methods, 106 semipermeable membrane, 442 SETTLE, 422 SHAKE, 420 Shannon, information, 455 shooting method, 80 SI units, xvii, xv simple Langevin equation, 267 simple pure Langevin equation, 267 simulated annealing, 205 simulated tempering, 205 Slater-type orbitals, 106 slow growth, in thermodynamic integration, 223 smart Monte Carlo, 273 SMD, see steering molecular dynamics smooth particle mesh Ewald, 168, 374 soft-core potentials, 225 solvated electron dynamics, 118 solvent accessible surface area, 352 SPC, see water SPC/E, see water speciﬁc heat isobaric, 426 isochoric, 426 spectrum, Python program, 329 SPH, see scaled particle hydrodynamics spherical harmonic functions, 361 spherical harmonics addition theorem, 361 spin glass, replica Monte Carlo, 205 spin matrices, 28 splines algorithms, 542 B-splines, 548 cardinal B-splines, 548 cubic, 526 Euler exponential, 553 ﬁtting, 530 for tabulation, 539 periodic, 529 quadratic, 526 quartic, 526 SPME, see smooth particle mesh Ewald SRstep, Python program, 87 standard hydrogen electrode, 440 standard reduction potential, 441 state functions deﬁnition, 425 extensive, 425 intensive, 425 table, 427 stationary state, 450 steady state, 450 steering molecular dynamics, 237 step disturbance, 508 Stirling approximation, 455 STO, see Slater-type orbitals stochastic equation mathematical incorrectness, 253 stochastic thermostat, 196 stoichiometric coeﬃcients, 437

Index Stokes’ law, 275 Stratonovich, 254 stress tensor and pressure, 292 coarse-grained, 292 ﬂuid dynamics, 283 Irving and Kirkwood localization, 484 localization, 483 quantum, 69 substantive derivative, 281 superatom approach, 249 superatom model, 249, 256 surface hopping, 113, 134 susceptibility electric, 336 magnetic, 336 switching function, 160 symmetry factor, statistics, 471 symplectic algorithm, 492 mapping, 492 notation, 492 transformation, 495 systematic force in Langevin dynamics, 251 temperature, SI unit of, xxiv temperature thermodynamic deﬁnition, 426 tempering, 205 ternary phase diagram, 434 thermal conductivity, from NEMD, 521 thermal wavelength, 469 thermodynamic cycles, 212 thermodynamic forces, 444 thermodynamic integration constraints, 226 deﬁnition, 223 for potential of mean force, 234 four-dimensional dynamics, 225 slow growth, 223 soft-core potentials, 225 vanishing particle, 226 thermodynamic perturbation, 235 thermodynamic potential excess, 219 particle insertion, 219 standard, 435 thermodynamics data sources, 444 degrees of freedom, 433 ﬁrst law, 429 second law, 424, 432 thermostat Andersen, 198 comparison, 203 extended system, 201 Gauss, 198 Nos´ e–Hoover, 201 stochastic, 196 weak-coupling, 199

tight-binding approximation, 107 time correlation function, 511 time, SI unit of, xxiv time-dependent DFT, 105 tin-foil boundary conditions, 166 TIP, see water TIP4P, see water torque, generalized, 402 torque, 405 total diﬀerential, 426 transformation canonical, 495 coordinate, 401 diagonalization, 385 Hamiltonian, 495 Jacobian, 495 of basis set, 384 symplectic, 495 unitary, 384 transition dipole moment, 36 transition rate multi-level system, 128 quantum dynamics, 124 two-level system, 126 translation, 403 triclinic box, 142 triple point, 433 Trotter formula, 387 Trotter–Suzuki splitting, 118 Trotter-Suzuki expansion, 193 truncated octahedron, 142 tunneling states, 82 two-form, 495 two-level system, 122 typographic conventions, xv umbrella sampling, 235 uncertainty relation, 22, 318 unitary transformation, 384 units atomic, xvii, xvii molecular, xviii, xvii rationalized, xvi SI, xv technical, xvi van ’t Hoﬀ equation, 442 van der Waals gas, 433 vanishing particle, in thermodynamic integration, 226 variational quantum Monte Carlo, 84 vector potential, 337 vector space complete, 381 complex, 380 deﬁnition, 380 n-dimensional, 380 normed, 380 real, 380 vector deﬁnition, 380

595

596 inner product, 381 normalized, 381 orthogonal pair, 381 orthonormal set, 381 representation, 379 scalar product, 381 velocity four vector, 26 velocity-Verlet algorithm, 192 Verlet algorithm, 191 vibration Morse function, 6 vibrational partition function, 472 vibrational-rotational-tunneling, 108 virial expansion, 436 virial theorem, 485, 504 virial Clausius, 485 periodic conditions, 486 virtual interaction sites, 157 viscosity bulk, 285 by NEMD, 520 from ﬂuctuation, 516 shear, 285 viscous stress tensor, 284 Voronoi tesselation, 306 VRT, see vibrational-rotational-tunneling walkers, Python program, 94 walkers, 89 water MCDHO model, 179 point-charge models, 179 quantum eﬀects, 41 relativistic correction, 30 second virial coeﬃcient, 175 SPC model, 179 SPC/E model, 179 TIP3 model, 179 TIP4P model, 179 wave four-vector, 26 wave group velocity, 24 weak-coupling thermostat, 199 wedge product, 494 Widom’s particle insertion, 219 Wiener process, 253 Wiener–Khinchin theorem, 327 Wiener–L´evy process, 253 Wigner correction, 72 Wigner rotation matrices, 407 Wood’s corrections, 168 Zeeman interaction, 34 zero-padding, 324

Index